text:
Motivation and emotion/Book/2011/Achievement motivation

"The roots of true achievement lie in the will to become the best that you can become. (Harold Taylor)"

Introduction
Have you ever wondered how some people go on to achieve great success in a chosen field, yet others seemingly do not have the same sense of achievement motivation? What drives them to excel? Can we influence or promote that same level of achievement in ourselves?

This chapter aims to give some insight into achievement motivation, and how we as students, teachers or parents can adopt strategies to encourage an achievement related mindset in ourselves and others, thereby satisfying one of our basic social needs for competence. Sufficed to say, there is no suggestion that changing your behaviours and habits will have you winning basketball games with the personal skill and mastery of Michael Jordan, but improvements to your own personal levels of talent in your chosen field are possible.

Definition and characteristics of high achievers
So what is achievement motivation? According to theorists, it involves desire for personal competence, competition with a level of excellence, in all likelihood imposed by parents with respect to culture (McClelland, Atkinson, Clark & Lowell, 1975). Indeed Michael Jordan showed extraordinary personal competence and desire in his chosen field of basketball. Rather than be branded merely genetically gifted, and borne out of his rejection from a high school basketball team, his extraordinary achievement motivation saw him labelled one of the hardest working athletes in the sport (Weiten, 2004). Likewise, when beginning his graduate education, B.F. Skinner used disciplined motivation in an attempt to compete with his fellow students (Myers, 2007). That is to say that Skinner scheduled his day to incorporate reading and study around labs, lunch and other daily activities, accounting for all but fifteen minutes in his day to foster his achievements in Psychology (Myers, 2007). Those high in achievement motivation show differences in their choice of task, latency, the amount of effort, persistence and willingness to take personal responsibility for performances when compared to those of low or lower achievement motivation (Atkinson & Birch, 1978; McClelland et al., 1975). Those high in achievement opt for tasks of moderate risk and probability of success, typically prefer feedback on their performance, and are more likely to delay gratification (McClelland, 1985). Whilst theorists have determined differences in these personal orientations, other factors may influence these outcomes.

Other influences in achievement motivation
McClelland et al. (1975) suggested that social and developmental influences, in particular parental child rearing practices, influenced high achievement motivation in children especially when parents adopted high standards. Moreover, children’s continual learning, cognitive development and the pleasure gained from progressive mastery cements personal competence and characterises high need for achievement (nAch) (McClelland, 1985). Motives are said to be based on incentives that are either positively or negatively arousing, thus producing an approach or avoidance of situations that fits into a particular cognitive schema (McClelland, 1985).

Does achievement motivation exist and if so, where's the evidence?
McClelland (1985) purports that tasks of moderate difficulty, competition and entrepreneurship are important in satisfying the need for achievement. Moderately difficult tasks give opportunities for the high achiever to gain more competence. McClelland surmises that as a result of these three situations, high achievers should do well in business as they adopt moderate risk, incorporate high personal performance and seek feedback. The nAch is usually measured by using the "Thematic Apperception Test" (TAT), a projective test that uses pictures of vague scenes. An individual's depiction of those scenes and the subsequent scoring of their answer is then said to reveal the individuals achievement motives (Atkinson & Feather, 1966). By examining the productivity and progress of entire countries, McClelland et al. (1975) have been able to use the TAT to test a society's overall achievement motivation.

Atkinson's model (classical view) - and dynamics-of-action model
This classical view of achievement motivation centres on motives as the central construct. Most notably in Atkinson’s model are the approach motive responsible for achievement motivation and the avoidance motive that encompasses the fear of failure (Atkinson & Birch, 1978; Atkinson & Feather, 1966; McClelland et al., 1975). Atkinson identified that achievement behaviour was dependent on
 * 1) nAch
 * 2) probability of success
 * 3) incentive for success.

Atkinson theorised that individuals exhibited a tendency to achieve (Ta), a tendency to approach success (Ts) with a tendency to avoid failure (Taf) (Atkinson & Feather, 1966). The tendency to approach success involved a number of processes including one’s motive for success, the perceived probability and the incentive value of that success (Atkinson & Feather, 1966). Likewise, within the tendency to avoid failure Atkinson believed that the motive to avoid, the perceived probability, and the negative incentive value for failure, all play a part in the overall achievement motivation (Atkinson & Birch, 1978; Atkinson & Feather, 1966; McClelland et al., 1975).

In terms of application of this theory, your tendency to obtain good grades will be dependent upon a number of things such as your motive to succeed, and the value you place on it. Conversely, if you don't require good grades (P's make degrees) and there is no motive to succeed, then expectancy and value of success will be decreased (Reeve, 2009).

In the dynamics-of-action model, Atkinson and Birch (1978) purport three causes are responsible for ongoing behaviour, namely -
 * 1) instigation - (Ts) - an approach tendency if an activity has intrinsic satisfaction;
 * 2) inhibition - (Taf) - an avoidance tendency if an activity has been thwarted or hindered in any way; and
 * 3) consummation - (conclusion of an activity) - an eventual reducing of that activity (Atkinson & Birch, 1978; Reeve, 2009).

For instance, according to the dynamics-of-action model, you are more likely to exhibit shorter latency and greater persistence in an achievement task if you were interested in that task and were motivated to do it, therefore showing less procrastination.

Achievement goals (contemporary view) - mastery/performance
In this contemporary view, Nicholls (1984) and Dweck (1986) theorised that two main achievement goals exist as the initial construct for this model. Mastery – which is the development of one’s competence, through self improvement, task progress, overcoming obstacles through sustained effort and persistence. With this definition of mastery - one can see the similarities in personal performances exhibited in the earlier example of B.F. Skinner and Michael Jordan. Ames and Archer (1988) identified a positive relationship between mastery goals and intrinsic motivation. Moreover, students that were oriented toward mastery goals reported more effective learning strategies, preference for challenging tasks, overall positive attitude and belief that success was relevant to effort (Ames & Archer, 1988; Klose, 2008).

The second goal orientation – performance goals are said to be concerned with one's ability or competence, the desire to display high ability while outperforming others, resulting in success with little apparent effort (Ames & Archer, 1988). Furthermore, when students identify performance as the significant goal, they linked failure to their own ability, hence promoting a non-encouraging self motivation (Ames & Archer, 1988).

Whilst both of these goals exhibit performance approach to certain tasks, adopting mastery goals rather than performance goals can produce outcomes whereby the individual will work harder, persist longer and ultimately perform better (intensity, latency, persistence). Individuals adopting a mastery goal are also more likely to ask for information and help, thereby increasing knowledge and understanding (Elliot & Church, 1997; Klose, 2008).

Integrated model
The integrated model is as the name suggests, a merging of the classical approach/avoidance model with the contemporary model of performance/mastery. The result of which sees an achievement goal concept or hierarchical model approach incorporating mastery, performance-approach, and performance-avoidance representations of underlying motive dispositions (Elliot & Church, 1997). Elliot (2006) defines approach motivation as a shifting or focusing of behaviour toward positive stimuli whilst avoidance motivation channels that behaviour by, or away from negative stimuli. The motive identified as being involved in mastery is achievement based, whereas performance avoidance's underlying motive is a fear of failure. Interestingly, Elliot and Church (1997) found that performance approach had both achievement and fear of failure as underlying motives, and this trend was also reflected in competence expectancies. Typically, performance avoidance goals indicated low competence and fear of failure.

The notion that competence arises from challenging stimuli that individuals are drawn to, is prevalent in Self-Determination Theory (Deci & Ryan, 1985). Furthermore, mastery goals are said to promote challenge, sustain self-determination and autonomy; whereas, performance (ego) goals are said to lower intrinsic motivation and produce higher anxiety (Deci & Ryan, 1985). Rawsthorne and Elliot (1999) conducted a meta-analytic review examining the effect of performance and mastery achievement goals on intrinsic motivation. They examined the literature for magnitude and direction of performance vs. mastery goals across studies, and they looked at inconsistencies across effects by examining possible moderators. Rawsthorne and Elliot obtained published literature from PsycINFO (American Psychological Association, 1971-1997) using keywords pertinent to achievement motivation, intrinsic motivation etc. Each study selected for review met criteria that assessed participants' self-reported achievement goals. Their findings suggested that performance goal oriented individuals produced diminished free choice, self report and interest when compared to those pursuing mastery goals, thus undermining intrinsic motivation. Scrutiny of moderators found that effects of comparison of performance and mastery goals were relient upon whether positive or negative feedback was received in reference to competence (Rawsthorne & Elliot, 1999). Furthermore, performance could also be directed toward a performance-approach or a performance-avoidance position when participants were encouraged to either a positive or negative performance outcome (Rawsthorne & Elliot, 1999). This meta-analysis shows that provided performance outcomes are positive, that is, the belief that one's performance is not fixed, but rather adaptable to change, then performance will increase.

What do those with high achievement motivation practice? So, how do I foster achievement motivation?
Dweck (1999) posits that personal qualities and characteristics can either be fixed and enduring (entity) or open to change which can be increased with effort (incremental). In effect, individuals are said to adopt goals in achievement motivation that are either entity or incrementally based (Dweck, 1999). Those of the entity belief will usually select performance goals, where fixed personal competence is shown; alternatively, incrementally focused individuals are more mastery focussed and will show changing levels of competence (Dweck, 1999; Howell & Buro, 2008). A study designed to examine the links between implicit theories and procrastination found that entity based beliefs were indeed positively correlated to procrastination (Howell & Buro, 2008). Given the fixed or changing competencies that implicit theory implies, Michael Jordan and B. F. Skinner are more likely to be incrementally focused with regard to their achievement motivation beliefs. Their persistence and intrinsic motivation and belief that they could always improve, shows that personal change is possible with effort.

Fostering achievement motivation in the student
"Competition is a by-product of productive work, not its goal. A creative man is motivated by the desire to achieve, not be the desire to beat others. (Ayn Rand)"

Student's intrinsic motivation to gain competence and mastery is strong in younger children but changes during ageing to be more extrinsic (Klose, 2008). It seems that self doubt becomes more common place whilst individuals try to place themselves socially. Indeed positive motivational orientation includes an emphasis on personal growth and mastery rather than the negative effects of believing that effort is futile if not compared with others (Klose, 2008). Students with a positive motivation see rewards as feedback for performance on a task, whereas negative motivation see reward and non-reward as feedback on their worth (Klose, 2008).

Klose (2008) proposes that fostering achievement motivation can lead to 'lifelong love of learning'. When mastery goals are adopted within the educational environment, students report their preference for challenging tasks and believe they use more effective learning strategies (Ames & Archer, 1988).

Kumar and Jagacinski (2011) propose that increasing task difficulty incrementally in the education environment is seen to have detrimental effects on some individual's perception of competence, especially those using a performance goal approach. They suggest that this decreases a student's perceived ability in their level of achievement. By decreasing the task difficulty, changes in their perceived level of achievement are likely to decrease changes in levels of perceived ability (Kumar & Jagacinski, 2011). Several studies have been carried out to determine whether success in the educational field is based solely on a performance goal approach, or indeed if other intrinsic factors may be present.

Predicting success in college study
"Think of yourself as on the threshold of unparalleled success. A whole clear, glorious life lies before you. Achieve! Achieve! (Andrew Carnegie)"

Are GPA's (grade point averages) (a performance goal approach) the only variable worth taking into consideration when determining who will eventually be successful? A study conducted by Harachiewicz, Barron, Tauer and Elliot (2002) examined this question in relation to high school students GPA's, achievement goals and ability to predict college success. Students from a first semester psychology class in a large midwestern university in America were selected for the study and tracked through to graduation. Achievement goals, interest and enjoyment of lectures were measured and final grades recorded and compared to admission grades and data. Findings for this study included the notion that interest (which represented commitment to the psychology field), was a strong predictor of achievement goals rather than ability and prior performance (Harachiewicz, et al., 2002). Having said this, there was a correlation between interest and grades having an influence on longer term academic choices, suggesting that performance goals also influence outcomes. Furthermore, performance-approach goals produced higher GPA's over the initial and overall college course however, mastery goals were found to be instrumental by fostering preliminary and ongoing interest in the course (Harachiewicz, et al., 2002). It is important to note that measurements from this study were conducted in a particular sample, which may not be representative of the general population and hence, could affect overall findings. Nevertheless, high school performance was a positive indicator for ongoing college success, but it didn't indicate interest. This and other findings suggest that modern achievement motivation theorists support a multiple goals perspective (Elliot & Church, 1997; Harachiewicz, et al., 2002) which is not only educationally based, other settings may support this perspective.

Fostering achievement motivation in sporting fields
"Achievement is not always success while reputed failure often is. It is honest endeavour, persistent effort to do the best possible under any and all circumstances. (Orison Swett Marden)"

Sport is important to us for a number of reasons including the social and developmental aspects and opportunities it affords (Deci & Ryan, 1985; Smith, Smoll & Cumming, 2009). As with educational settings, sport provides similar approach goal patterns of thought, that is that mastery is associated with dedication, effort, intrinsic motivation; ego-oriented individuals on the other hand, link success with superior skill (Smith et al., 2009). Individuals who are mastery oriented report of greater enjoyment of the activity with lower pre event anxiety levels when compared with ego driven individuals (Smith et al., 2009). Whilst ego-driven performance is associated with high achievement, it can produce inconsistent performance and an overall decrease in intrinsic motivation (Deci & Ryan, 1985; Smith et al., 2009).

Motivational climate is referred to as a setting that is conducive to parents, teachers, coaches promoting self-improvement, task mastery, persistence and effort (Klose, 2008; Ames & Archer, 1988) whereas, ego driven climates promote socially comparative settings emphasising winning over improvement (Smith et al., 2009). Sport itself may offer a motivational climate that is more easily recognisable than those in educational settings. For instance, Smith et al., (2009) assert that effort and persistence can be observed through physical signs such as sweating, increase in breathing rate etc. when performing a task. Effort and persistence are not always as easy to detect in the classroom setting (Smith et al., 2009). Empirical evidence identifies coaches as major influencers in promoting mastery in athletes (Smith et al., 2009), therefore, the obvious visual cues produced by athletes may be a useful tool in the coaches arsenal to showcase effort and persistence. With regard to individual wellbeing, athletes higher in mastery report higher levels of fun when compared to those athletes whose coaches promoted winning or ego based activity (Smith et al., 2009).

Checklist for coaches
 * provide a mastery climate,
 * promote teaching in skill improvement over winning (promotes wellbeing),
 * showcase effort and persistence,
 * remember self-improvement sustains greater happiness and fun over individual wins/competition; and
 * encourage incremental over entity theory - emphasise the value in effort.

Summary
Our desire for personal competence and competition with a standard of excellence is in essence, our achievement motivation. It can be influenced by parents, coaches, educators and/or by adopting strategies that promote mastery and approach goals. It is imperative that the value of effort is promoted - as in incremental over entity theories. Of particular importance is the value we place on environments that adopt positive feedback and endorse individual effort. We can achieve this by -

 * rewarding performance on specific tasks that show self improvement and effort;
 * not comparing individuals to siblings or others;
 * discussing motivation and individual progress with students/athletes/individuals; and
 * above all protecting the self-worth of individuals.

With sustained effort and persistence, high competence expectancies and intrinsic motivation, individuals will be bringing out their own 'Michael Jordan' or indeed the 'B. F. Skinner' in themselves.

Assess what you have learned
{Those high in achievement motivation tend to work for personal fulfillment rather than reward. + TRUE. - FALSE.
 * type=""}

{External rewards are better than intrinsic motivators for fostering achievement motivation. - TRUE. + FALSE.
 * type=""}

{Those high in achievement motivation select moderate levels of challenge rather than very easy or very difficult tasks. + TRUE. - FALSE.
 * type=""}

{Providing a competitive environment is better for fostering high need achievers than promoting mastery. - TRUE. + FALSE.
 * type=""}


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Motivation and emotion/Book/2011/Negative thinking and emotion

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Introduction
Positive psychology has become increasing popular, it is for this reason it is important not to revert to the one size fits all approach because we are all unique. Negative thinking is a natural occurrence yet it is known to carry a negative stigma. Surprisingly enough, negative thinking, also known as pessimism has been known to increase personal growth and performance (Norem and Chang, 2002). Negative thinking is a defensive mechanism to prevent us from going down the wrong path. However, at times negative thinking can become destructive to your psyche leading to depression, if those thoughts are seen as facts rather than a caution. Negative thinking should be incorporated into emotion regulation in which you filter how you feel. Depression is defined as fear of the future based on one's past; where negative thoughts from previous incidents forecast a gloomy realistic outcome. It is for this reason it is important to acknowledge negative thoughts but not to rely on them as facts. This important fact will change the way you feel (Chang, E.C, Norem, J.K ,2002). There are times when pessimism and negative thinking are indeed positive psychology, as they lead to better performance and personal growth Freud believed we subconsciously ignore negative thoughts so as not to affect our egos and emotions. Even though these negative thoughts can lead to depression if not controlled they help us survive.

Defensive pessimism
Although there is substantial empirical evidence that optimism leads to better coping strategies, wellbeing and satisfaction there is also proof that pessimism can improve your lifestyle as well. Pessimism consists of numerous subgroups some more helpful than others. Such subgroups include: unrealistic pessimism, neurotic and rational pessimism, defensive pessimism and strategic pessimism.

Defensive pessimism is linked to more positive outcomes than dispositional pessimism. Defensive pessimism is a well used strategy anxious people use to obtain their goals. These individuals will set low unrealistic expectations and then reflect on all possible outcomes associated with that goal. It was found by Cantor & Norem, (1989) that defensive pessimists perform just as well as strategic optimists if not better as both strategies decreased anxiety in anxious individuals. This demonstrates that negative thinking (pessimism) does not necessarily hinder ones existence.

Although defensive pessimism allows you to tackle obstacles while decreasing anxiety, not all forms of pessimism are supportive. Unrealistic pessimism was found to hinder one’s lifestyle, preventing them from achieving goals they decide to pursue. This is when one exaggerates a scenario and imagines every possible negative outcome in that scenario. E.g scenario: If I go swimming a shark will swim next to me. Outcomes: I will die, it will attack me, I will be scared. Result: I will not go swimming.

People who listen to their negative thoughts are more likely to construct safety plans preventing disasters when they occur. For example, you decide to go on holiday and your negative thoughts convince you to apply for travel insurance. Imagine the relief felt when a disaster strikes while you're overseas but you are completely covered by insurance. As they say there is a silver lining to every cloud.

The reason why negative thinking is so effective is because it is normally unconscious thinking. The capacity principle demonstrates that unconscious thoughts have a higher capacity of consciousness. This is because consciousness only uses a subset of information and impedes the quality of choice Negative thinking can be explained using the unconscious thought theory in which both conscious and unconscious thoughts impact on the choices we make. Negative thinking is normally the part the unconscious thinking occurs when one is unaware and impacts on a person's behaviour. This is normally at its most influential when one is distracted. This is known as the unconscious thought principle (Bos, Baaren, Dijksterhuis, 2008).

Cognition Model of Depression
Even though negative thinking is important to achieve our goals when it becomes ruminative and cycles endlessly it may lead to serious mental illnesses like depression. Teasdale (1983) found that depressed clients easily retrieved negative memories (thoughts) but struggled to retrieve positive memories during a word association test. By using the latency of retrieval as a measure of accessibility, depression was found to increase accessibility of negative thoughts and decreased accessibility of positive memories. Negative cognitions can produce and maintain a state of depression by combining continuous unrealistic negative thinking with the chemical imbalance in depression. Beck’s (1967) cognitive models reported that negative cognitions can have a causal role in producing symptoms of depression. Before, it was thought that negative thinking was thought to be a symptom of depression however now it is known to actually a causal factor or depression.

The cognition model of depression suggests a person’s pessimism, low self esteem and negative outlook leads to depression by biasing ones thinking in a negative direction. It becomes a hard habit to break when the negative thinking process is established within the brain. This is why it’s important to break this cycle. Another possibility the cognition model states is that negative thinking produces and maintains a state of depression whilst bias thinking, meaning that both depression and cognition have a reciprocal relationship.

Beck (1967) reported depression to be associated with increased levels of negative thinking. When these thoughts were increased a deeper level in depression resulted, and decreasing negative thinking alleviated depression. They also found through positive experiences one became more confident and decreased negative cognition thus alleviating depression. Another therapy that proved to be successful is the power of distraction which also decreased depression.

As noted previously negative thinking is essential in our progress only when balanced. When negative thinking becomes a truth rather than possibility your emotions become compromised. Depression is essentially negative thinking which then manifests into negative effect. Excessive negative thinking can be known as pessimistic thinking, this would have an obvious upsetting, depressive affect upon ones emotion ( Ball, S.A., & Zuckerman, M, 1990).Unfortunately negative thinking can consist of unconscious thoughts and easily converts into the strongest of habits. People easily generate negative thoughts automatically and are unaware of its occurrence. So when you start noticing constant sadness and feeling of doom, perhaps it is time to find a balance to this delicate process.

Black Glasses
It is important to understand why negative thinking may impact ones emotions one answer is the black glasses theory. People with low self-esteem hardly acknowledge positive experiences; as they do not weigh as much negative experiences. At the same time they fixate on their weaknesses instead of deliberately concentrating on their strengths (Sanna, L.J ,1996). A good example is the prosecutor who’ll call your attention to every mistake or any weakness presented without pointing out any positive aspects this encourages negative thinking in way which will negatively impact your emotion and behaviour.

Another affect negative thinking can have on your emotions is anxiety. Anxiety can be defined as fear on the possible future, where negative thoughts rule and dictate what WILL occur in your future. This is why it is incredibly important to acknowledge the possibility of a negative thought but to realize it is not the truth, just a thought. Again, there is a reason for this brain process. Put simply, the more anxious one is the more likely you will take extra precautions to ensure your survival. In other words unconscious negative thoughts can improve one's decisions by exhausting more possibilities. Anxiety can assimilate a lot of stress and unhappiness in your life that is why it is important to find a balance between negative thoughts and how they affect your emotions.

My Worst Nightmare Came True
Negative thinking when it is not controlled can not only lead to emotional imbalances, such as depression and anxiety, it can lead to your demise; The self-fulfilling prophecy is when ones negative thinking becomes a reality and it normally correlated with ones worst fear associated with their emotions. The self-fulfilling prophecy is a psychology theory defines by Robert Merton as: when a false definition of a situation creates a new behavior which generates the initial false conception to be true. In other words because you begin acting a certain way because of your negative thoughts, the path you were trying to avoid, presents itself (Dolinski, D., Gromski, W., & Zawisza, E. ,1987).

Negative thinking effect your emotions this way due to the weighting principle in which unconscious thoughts automatically weight factors in a situation which you may not be consciously aware of. When a negative thought presents itself, you begin to reflect on it and it negatively effects your mood. This will change your behaviour which normally reinforces the likelihood of the imagined negative outcome occurring. The end result is the Self-fulling prophecy in which negative thinking about a worst case scenario reinforces your behaviour towards that direction; making they worse case scenario a reality. E.g. my husband is going to cheat on me, behaviour: act suspicious, don't trust him, outcome: husband begins to think about it, or cheats because wife doesn't trust him anyway.

Finding the Balance
As hard as it is to believe everyone experiences negative thinking. A lot of us create a facade in which we are invulnerable and burst with confidence. However, underneath our emotions are susceptible to different thoughts especially negative ones.

The Differential self-awareness theory explains how we can be privately aware and publicly aware (Affleck, G., Tennen, H., & Apter, A. ,2001).

Private self-awareness refers to a person who is looking within himself or is thinking about himself, and being aware of the hidden aspects of himself.

Negative thinking can affect us both ways depending on the situation and our identity. The prominent key is to remain sceptical without being overwhelmed by these thoughts. Visionaries like Bill Gates and other successful individuals found negative thinking to provide them with an advantage over their competitors, however in order to resolve a problem introduced with a negative thought it is important not become overwhelmed and be able to differentiate reality from possibility.

Breathe in Breathe Out
One simple but incredibly essential method to controlling your negative thinking is your breathing. Negative thinking can leave a feeling of discomfort and discontent in one self. Some familiar feelings resemble a heavy, compounding mass in the center of one’s chest. Although this sensation cannot be removed without removing reason for the negative thought it can be controlled (Kelman, H., 1945). Using this simple ritual next time you feel overwhelmed and the heavy mass in your chest. try taking a deep breath in, visualizing the area around the heavy dark mass expanding, giving it more room, and as you exhale visualize your expiration blowing around the mass. Although the emotion cannot be removed it can be controlled.

Power of Imagination
Another useful technique in impeding the negative thought process in negative thinking is visualization. Cyclic negative thoughts become we established thought process with high accessibility. A way to stop this is imagination when you think badly of yourself an exaggerated version of that thought. E.g I am stupid, imagine yourself wearing clown shoes, a stupid outfit to the point it make you laugh. Then imagine an exaggerate form of yourself being brilliant looking like superman. The last step is first imagining yourself as a stupid figure everyone laughing at you, and then stripping all that silliness away to reveal the brilliant person of yourself. This way you are not stopping the thought processes which will just intensify the thoughts but you are changing the direction of your thoughts. Another well known technique is to imagine a bright powerful sun piercing through the negative thought every time you think pessimistically. You will begin to notice how many of these thoughts you encounter each day which stripes the power of them.

Conclusion
Although negative thinking are unconscious thoughts that can be used to solve logical problems when they are uncontrolled, they can be detrimental to ones behaviours and emotion. That is why it is important only to acknowledge them without necessarily believing them. This exceptionally applicable method to control negative thinking is by changing the way you think People do not have much control over their emotions however they have the capability to manipulate the way they think. Every time negative thought appears which would be detrimental to your emotions visualize that it is only a thought, not fact. Although thoughts are important they cannot dictate your actions if you do not let them. Understanding that they are merely possibilities instead of facts, will deeply improve your emotions and create stability in your life.

Negative thinking can either benefit your emotions or impede them depending on the way you react to them and how much you weight them. Numerous factors will determine this such as: its weighting, your current self-esteem, the brain process involved and your current situation. The weighting depends on whether the thought is brought upon on a subconscious level which is normally the case; however people who have a depressive nature will normally encourage this type of thinking at a conscious level. Things brings upon a whole other set of criteria as conscious thoughts are processed and weigh differently to unconscious thoughts (Canter, N, & Norem, J.k., 1989).

It has been established that negative thinking is a defence mechanism which guides us towards the right decisions we wish to make, by providing us with more possibilities and more realistic future. This however stops’ being helpful when one refuses to differentiate reality from possibility. Many different techniques are available to support a healthy balance between negative thinking and emotions one of which is to become privately aware of your thinking. Unconscious thoughts normally slide through undetected when we are distracted or stressed. That is why it is important to remain focused and try catch” yourself in the action (Teasdale. J.D & Russell. M.L., 1983). Another main technique is to focus on your breathing to bring you back to the present from the future thinking. You will begin to realize all is not so bad.

Finally, low self-esteem is a main contributor to negative thinking becoming a disability rather than a skill. Therefore improving one’s self esteem will not only be beneficial to one’s mental health and social life but it will also impede the easily established habit of hurtful negative thinking.

Perhaps the best saying to remember when caught in the negative process is: Just because I’m thinking it doesn’t mean it’s true.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
Motivation and emotion/Book/2011/Grief

Overview

"To spare oneself from grief at all cost can be achieved only at the price of total detachment, which excludes the ability to experience happiness - Erich Fromm"

Grief is extremely stressful and can have a massive impact on an individual’s daily functioning. Grief is common and it is likely that you will experience grief if you haven’t already or grief will affect someone close to you. Knowing how to effectively deal with personal grief and the grief of others can be difficult. The present chapter aims to provide a comprehensive understanding of grief, how to effectively manage personal grief and the grief of others.

What is grief?

Grief is universal phenomenon that a can be thought of as any form of distress in response to loss (Howarth, 2011a). Grief is complex as it involves emotional, cognitive, physiological and behavioural manifestations. Grief is universal but its symptoms are not (Howarth, 2011a). People grieve for different lengths of time and in different ways as grief encompasses many emotions and its expression is unique to the individual (Howarth, 2011a). Grief effects people of all cultures and has been found to consistently effect people throughout history (Granek, 2010).

Often the words grief, bereavement and mourning are used interchangeably but they do have different meanings. Bereavement is grieving in response to the death of someone significant. Mourning refers to public displays of grief (Granek, 2010). Mourning is different cross culturally as it relates to cultural practices through which grief and bereavement are expressed (Stroebe, Hansson, Schut, & Stroebe, 2008). Mourning is related to the customs of a society as it is influenced by social context. Differences in mourning customs between cultures can be seen in religious funeral services (Stroebe et al., 2008).

Bowlby

John Bowlby is most recognized for his work on attachment theory and has used attachment theory to provide a biological explanation of grief (Stroebe, 2002). Attachment can be defined as a meaningful bond that is made with a significant something or someone and is important for maintaining relationships (Field, 2006). Bowlby thought that people formed attachments for protection and survival. An attachment relationship that serves a protection and survival function can be seen in the attachments made between mother and child (Mikulincer & Shaver, 2008).

Bowlby theorised that when someone feels threatened or distressed their attachment behavioural system is activated and they will seek protection and comfort from those who they have formed an attachment with. When a significant attachment figure dies or is lost the individual will experience extreme distress which Bowlby called separation distress (Mikulincer & Shaver, 2008). The individual fears life without their attachment figure as they can no longer receive support from them. The individual will experience grief until they become detached from their former attachment. According to Bowlby reorganisation happens when the individual accepts the loss and integrates the loss into their future (Mikulincer & Shaver, 2008). This model of grief has been criticised as not all individuals grieving styles follow distinct stages however stages are useful in informing people what is involved in the grieving process (Mikulincer & Shaver, 2008).

Kübler-Ross
Swiss psychiatrist Elizabeth Kübler-Ross proposed a stage theory of grief from her work with terminally ill hospital patients (Krueger, 2006). Kübler-Ross’ theory of grief was used to explain the process of grief in terminally ill patients and their loved ones reactions to loss (Krueger, 2006). Kübler-Ross’ theory consists of five stages: denial, anger, bargaining, depression and acceptance. The theory has been criticised as it comes from Kübler-Ross’ Subjective experiences with dying people and the theory does not take into account individual differences (Retsinas, 1988).

Denial – when people first face loss they enter a state of denial which may last for months. In this stage of Kübler-Ross’ theory people are in shock and find it easier to deny the reality that they have a terminal illness (Bolden, 2007).

Anger – after denial comes anger. The grieving person becomes extremely angry about anything and everything as they may feel it is unfair that they have been singled out. this stage can be difficult for carers or loved ones of the grieving person as they can become a target of the anger (Bolden, 2007).

Bargaining – once the grieving person has passed the anger stage they enter a bargaining stage. Within this stage the individual realises that death is inevitable but pleads for more time and feels that they would do anything for more time. The individual bargains with doctors, their god or just with themselves. It is within this stage that any unfinished business or loose ends are dealt with (Bolden, 2007).

Depression – the individual becomes depressed after they have taken care of unfinished business as they are starting to accept the inedibility of the loss. it is thought that the individual will experience reactive depression which refers to grieving losses that have already been experienced. After reactive depression comes proactive depression which is the grieving of future losses (Bolden, 2007).

Acceptance – the last stage of Kübler-Ross’ theory is acceptance. The individual becomes detached and accepts the inevitability of death (Bolden, 2007).

Trajectories of grief
A study was conducted by Bonanno et al. (2002) that aimed to identify the most common trajectories of grieving. Bonanno et al. used data from previous studies that had interviewed widows before and after their partners had died. Bonanno et al (2002) found five trajectories of grief.

 Common grief  – these people have low preloss depression, high postloss depression 6 months after loss but recover at 18 months after the loss (Bonanno et al., 2002).

Stable low distress  – these people have low preloss depression and low postloss depression at 6 and 18 months after the loss (Bonanno et al., 2002).

Depression then improvement  – these people have high preloss depression but low postloss depression at 6 and 18 months after the loss (Bonnano et al., 2002).

Chronic grief  – these people showed low preloss depression and high postloss depression at 6 and 18 months after the loss (Bonanno et al., 2002)

Chronic depression – these people have high preloss depression and high postloss depression 6 and 18 months after the loss (Bonanno et al., 2002).

Physiology and grief

Research by Kowalski and Bondmass (2008) have found that grief causes significant physiological change. Participants of the study were 173 widowed women who completed questionnaires relating to their grief. The women were asked if they had experienced any physical symptoms after the death of their partner. Results of the study found that the women reported having experienced pain, trouble sleeping, symptoms that required surgery, gastro intestinal issues and neural and circulatory problems (Kowalski & Bondmass, 2008). A similar study by Fogel (2007) found that grief and depression affect cardiovascular health.

Normal grief

Grief involves complex emotions that affect people differently. Grief is subject to cultural variation and at times can be difficult to distinguish from Major Depressive Disorder (Howarth, 2011a). This makes it hard to define what normal grief is, but a defining feature of normal grief is reconciliation (Doughty, 2009). Reconciliation is achieved when an individual no longer grieves a loss and can move on with their life. To achieve reconciliation, it is thought that people must accept the loss, fully grieve the loss, make adjustments to their life, incorporate the loss into their self-concept, acknowledge that what was lost will not be a part of their future but rather a memory, find meaning from the loss to be able to move on and make new bonds in life (Doughty, 2009).

As grief affects people differently research suggests that there may be different styles of grieving that are shaped by personality, gender and culture (Doughty, 2009). Research proposes that individuals have adaptive grieving styles that influence their behaviour, cognitions and strategies used to deal with loss. Adaptive grieving styles are distinguished by how an individual outwardly expresses a loss and how a loss is experienced internally (Doughty, 2009). According to the adapting grieving styles theory, an individual’s grieving style can range between intuitive and instrumental grieving (Doka, 2002).

Intuitive grief refers to a greater expression of emotion as a result of the loss and a desire to talk about the loss. Intuitive griever’s outward expressions reflect their inner feelings (Doughty, 2009). Intuitive grievers feel sad and express their sadness through crying and talking about their loss. Intuitive grief is stereotyped as a female grieving style (Doka, 2002). Instrumental grievers do not display their emotions as intuitive grievers do because they do not feel the need to (Doughty, 2009). This may be because their grief is not experienced as intensely as an intuitive griever experiences grief. Instrumental grievers aim to control their emotions and may see their grief as a problem solving challenge (Doughty, 2009). People with an instrumental grieving style appear unaffected by loss as they do not express grieving related behaviours such as crying (Doughty, 2009). Instrumental grief is stereotyped as a male grieving style (Doka, 2002). The most common adaptive grieving style is a blended style that incorporates both intuitive and instrumental grief (Doughty, 2009).

Complicated grief
Complicated grief is experienced by people who are unable to overcome grief (Strada, 2011). Their grief response is considered abnormal as it is persistent in its intensity and duration. It is thought that complicated grief arises as a result of deficits in emotional regulation (Gupta & Bonanno, 2011). The symptoms of complicated grief are similar to Major Depressive Disorder symptoms but can be distinguished by an individual’s extreme longing for what is lost, strong disbelief of the loss and feelings of loneliness and detachment (Strada, 2011). People with complicated grief also feel that their life no longer has meaning after loss (Strada, 2011). As complicated grief is different from Major Depressive Disorder it will be included in the next edition of the Diagnostic and Statistical Manual of mental Disorders as Prolonged Grief (Strada, 2011).

Disenfranchised grief
Disenfranchised grief is when a person is denied their right to grieve. The individual still experiences grief but it is not socially acknowledged or validated (Doka, 2002). The grief is not acknowledged for many reasons, such as the nature of the relationship to what is lost is not socially acceptable, or the manner in which the individual mourns is not socially acceptable (Doka, 2002). Loss that is not socially acknowledged can include the death of a pet, the death of a unborn baby, a missing family member or friend, losing a job, having a family member on death row or losing property in a fire (Attig, 2004). When an individual’s grief is disenfranchised, society denies them a grieving role which limits their support networks and they are denied compensations such as time off from work (Doka, 2002).

It is important to understand that people can become attached to many things and any form of loss can cause someone to grieve. To help someone recover from loss, their grief needs to be recognised and acknowledged (Attig, 2004). Being empathetic and respectful towards someone’s loss is ethical and prevents their grief from becoming disenfranchised (Attig, 2004).

 The relationship is not recognised – most societies consider family relationships to be the most important. Friendships and attachments to material possessions are considered less important (Doka, 2002).

The loss is not acknowledged – societal rules decide what losses are significant and what losses are not (Doka, 2002).

The griever is excluded – sometimes people are socially defined as incapable of grief. This can be seen in situations where a person has a disability or mental illness and is excluded from grieving rituals such as planning funeral services because their grief is not acknowledged (Doka, 2002).

Circumstances of the death – the way in which a loved one dies can disenfranchise grief. People who lose a significant person in their life to AIDS, homicide or suicide feel stigma which prevents them from receiving full support (Doka, 2002).

The way an individual grieves – grief effects people differently. Support is more likely to be given to people who have strong immediate reactions in response to loss than people who grieve later (Doka, 2002).

Anticipatory grief
Anticipatory grief is grief that results from the knowledge of an impending loss (Cheng, Lo, Chan, Kwan & Woo, 2010). Grieving prior to a loss can be experienced by the carers of loved ones with a terminal illness or people who have a terminal illness (Cheng et al., 2010). Anticipatory grief refers to any loss that is anticipated, not only to anticipating death from terminal illness. Yet most of the research available on anticipatory grief involves people who are terminally ill and the effect anticipatory grief has on their families (Cheng et al., 2010).

Research has found that terminally ill cancer patients deal with many losses that are associated with anticipatory grief when they learn that they only have a certain time left to live (Cheng et al., 2010). Their losses include the loss of a future with friends and family, loss of hopes and dreams for the future, loss of cognitive functioning, loss of identity and role definition and the loss of independence as their cancer worsens (Cheng et al., 2010). Carers of people with terminal illnesses experience anticipatory grief as they fear the eventual death of their loved one but also experience loss related to the kind of illness their loved one has died from (Holley & Mast, 2010). Carers of people with dementia grieve the loss of the relationship that they had with the dementia sufferer as dementia symptoms worsen and affect cognitive and physical abilities (Holley & Mast, 2010). Caregiver’s anticipatory grief is also associated with the sacrifices, burdens and isolation that come with caring for someone with a terminal illness (Al-Gamal & Long, 2010).

Grief in childhood and adolescence
Children experience grief but experience it differently to adults. Depending on the age of the child they may not understand that death is forever (Willis, 2002). As children develop they learn that some things are lost forever, can grieve previous losses that were not originally understood (Willis, 2002). Children grieve in bursts that are expressed behaviourally and emotionally. For children to effectively recover from loss they need a safe and stable environment and an adult who they can depend on (Himebauch, Arnold & May, 2008).

0-2 years – children under 2 years of age do not understand death and loss but do experience grief in situations such as when they are separated from their parents or carers (Himebauch et al., 2008). in accordance with Bowlby's attachment theory, when infants are separated from their parents or carers they will experience separation distress.

2-6 years – preschool aged children do not understand the permanency of death and loss. They do feel grief when they lose something close to them and will often feel that the loss was their fault. It is important to let the child know that they did not cause the loss and provide them with factual explanations (Himebauch et al., 2008).

6-8 years – primary school aged children know that death is permanent. They experience grief and may fear death. To help the child cope with their grief school teachers should be informed of the loss (Himebauch et al., 2008).

8-12 years – teenagers understand death as adults do and grieve in much the same way but may have trouble expressing their feelings. It is not uncommon for teenager to engage in risky behaviour to challenge their mortality. Grieving teenagers need freedom to spend time with their friends and supportive environment (Himebauch et al., 2008).

Managing and overcoming grief
Grief counselling is a method used to alleviate peoples grieving and aims to help people grow from the traumatic experiences caused by grief (Altmaier, 2011). Most people will overcome their grief in time but for those whose grief appears to be chronic or for people with anticipatory grief, grief counselling is a useful treatment. It is thought that grief counselling creates a sense of hope for the future in the grieving client (Cutcliffe, 2006). By experiencing a caring human connection with the counsellor the client is able participate in a positive relationship. The counselling relationship is a stable relationship that provides the client with consistent support (Cutcliffe, 2006).

Another approach to managing grief could be writing about it. Research by Lichtenthal and Cruess (2010) studied the effect that writing about loss has on grief. The grieving participants in this study wrote for 20 minutes about their grief. Their writing focused on trying to make sense of their grief and trying to identify the good aspects of their life (Lichtenal & Cruess, 2010). The results found that spending time writing about the good aspects of their life after loss and trying to make sense of personal grief was effective in helping people adjust to life after loss (Lichtenal & Cruess, 2010).

For children to overcome grief they must be able to accept that what they have lost is gone forever. For this to happen they need to be informed of the facts surrounding the loss (Howarth, 2011b). Children need to be given time to grieve the loss and experience the pain that is associated with the knowledge that the loss is permanent. After children have grieved the loss the feelings of grief becomes less frequent and they can learn to how to live without what was lost and convert the loss into a memory rather than something that remains present (Howarth, 2011b). Specific factors that influence how children overcome and manage grief is the child’s age, the safety of their environment and stability and support of their caregivers. Children manage their grief better when they are supported emotionally by their caregivers and when they feel safe in their environment. If children grieve as a result of the death of someone close to them, they should be allowed to participate in rituals such as funeral services to help them overcome their grief (Howarth, 2011b).

Conclusion

 * Grief is any distress in response to loss.
 * Grief is universal and effects people differently.
 * The five most common trajectories of grief are: Common grief, Stable low distress, Depression then improvement, chronic grief and Chronic depression.
 * It is thought that that people grieve because they experience separation distress when they can no longer make contact with what is lost.
 * When faced with loss people will pass through the stages of denial, anger, bargaining, depression and acceptance.
 * Grief causes significant physiological change and has been found to affect physiological health.
 * Different types of grief include; normal grief, complicated grief, disenfranchised grief and anticipatory grief.
 * Normal grief involves reconciliation and recovery from grief.
 * Complicated grief is experienced by people who are unable to overcome grief.
 * Disenfranchised grief is when a person is denied their right to grieve.
 * Anticipatory grief results from the knowledge of an impending loss.
 * Children grieve differently to adults. 2 year old do not understand loss, 2-6 year olds do not understand the permanency of death and loss, 6-8 year olds know that death is permanent and 8-12 year olds understand the permanency of death and grieve in much the same was as adults do.
 * Ways to manage and overcome grief include grief counselling and writing about loss.
 * Grieving children need a safe environment and support from their caregivers to be able to overcome grief effectively.

Quiz
{The stage theory of grief that involves the stages: denial, anger, bargaining, depression and acceptance was proposed by? - Bowlby. - Bonanno. + Kübler-Ross.
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{Anticipatory grief is grief that results from the knowledge of an impending loss. + True. - False.
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{People grieve for different lengths of time and in different ways. + True. - False.
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{preshcool aged children understand death as permanent. - True. + False.
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title: Motivation and emotion/Book/2011/Grief


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User:Egm6936.f10/Probability concepts

=Probability concepts and notations=

djvu notes: [[media:vql.probability.cdf.djvu|Probability, distribution, density]]

Events(Samples, Outcomes)
Event (or sample or outcome) is a subset of results of random experiments, which designated by $$\mathbf\omega $$.

For example, the result of tossing a coin once should be either head or tail, i.e., $$\mathbf\omega =$$ head or $$\mathbf\omega =$$ tail. If we toss the coin twice, then $$\mathbf\omega =$$ {head, tail} or $$\mathbf\omega =$$ {head head} or $$\mathbf\omega =$$ {tail head} or $$\mathbf\omega =$$ {tail tail}.

Algebra of Events
In some ways, the algebra of events share some similarities with the algebra of real numbers, with intersection ( $$\cap$$ ) corresponding to multiplication ( $$\times $$ ), complement ( $$^{c}$$ ) to subtraction ( $$\mathbf -$$ or $$ \setminus $$ ) and union ( $$\cup$$ ) to addition ( $$\mathbf +$$ ).

The union of $$n$$ events $$ A_{1},\, A_{2},\, ..., A_{n} $$ is the set collecting all points in all those events $$ A_{1},\, A_{2},\, ..., A_{n} $$.

Notation:

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$$\displaystyle A_{1} \cup A_{2} \cup ... \cup A_{n}$$, or $$\bigcup_{i=1}^{\infty}A_{i} $$ (1)
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The intersection of $$n$$ events $$ A_{1},\, A_{2},\, ..., A_{n} $$ is the set collecting all points belonging to all those events $$ A_{1},\, A_{2},\, ..., A_{n} $$.

We call they are disjoint if the intersection of sets is empty.

Intersection has the associative property.

Notation:

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$$\displaystyle A_{1} \cap A_{2} \cap ... \cap A_{n} $$, or $$\bigcap_{i=1}^{\infty}A_{i} $$ (2)
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$$\displaystyle (A_{1} \cap A_{2}) \cap A_{3} $$=$$ A_{1} \cap (A_{2} \cap A_{3}) $$ (3)
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The complement of a event $$A$$ in $$\Omega$$ is the set collecting all points in $$\Omega$$ but not in the event $$A$$. Generally, we can have two kinds of complement: relative complement and absolute complement.

Relative complement of $$A_{1}$$ with respect to $$A_{2}$$ is the set of points in $$A_{2}$$ but not in $$A_{1}$$. If union of all sets $$A_{1},\, A_{2},\, ..., A_{n}$$ considered to be $$\cup$$, the absolute complement of $$A_{1}$$ is the set of points in $$\cup$$, but not in $$A_{1}$$.

Notation:
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$$\displaystyle A^{c} $$ (4)
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For schematic representations of union, intersection and complement, we can use Venn Diagram.

Fig 1: Venn Diagram

Sample space (Outcome space)
Sample space(Statistical Theory) or outcome space(Probability Theory) is a collection of all possible outcomes (or events or samples) of random experiments, which denoted by $$\mathbf \Omega$$.

For example, in the coin - tossing experiment, a coin is tossed once, the outcome space $$\mathbf \Omega$$ ={heads, tails}. If we tossing twice, the outcome space $$\mathbf\Omega =$$ { {head, tail}, {head head}, {tail head}, {tail tail} }.

( Xiu 2010, p.9 ;, Shao 2007, p.1 .).

Sigma-Field
Sigma-Field is a collection of subsets of a sample space $$\mathbf\Omega$$ (not necessary all), which denoted by $$\mathcal F$$. For instance, $$\mathcal F = \{ \emptyset, {\rm heads}, {\rm tails}, \mathbf\Omega\}$$ in the coin - tossing experiment.

Three conditions that the sigma-field must satisfy:

$$\bullet$$ Non-empty: $$\Omega \in \mathcal F$$ and $$\emptyset \in \mathcal F$$;

$$\bullet$$ Given $$A \in \mathcal F$$, then $$A^c \in \mathcal F$$;

$$\bullet$$ Given $$A_1$$, $$A_2$$,...$$\in \mathcal F$$, then

$$\bigcap_{i=1}^{\infty} A_{i} \in \mathcal F$$ and $$\bigcup_{i=1}^{\infty} A_{i} \in \mathcal F$$.

i.e., sum or union of any subsets of $$\mathcal F$$ is a subset of $$\mathcal F$$.

( Xiu 2010, p.10 ;, Shao 2007, p.2 .)

Note: $$\mathcal F$$ is called a " sigma-field " or " sigma-algebra ", written as $$\sigma$$-field  or  $$\sigma$$-algebra. $$\sigma$$ is mnemonic for " S ", and " Sum ", due to property.

Probability
Probability is used to measure the likelihood of the occurrence of certain event (or outcome). Probability of an event $$\mathbf \omega $$ belonging to an element $$A \in \mathcal F$$ is a non-negative number (or measure), which is mathematically denoted by

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$$\displaystyle P(\omega \in A)=P(A) $$ (5)
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For example, in the coin - tossing experiment, $$ P(heads)=P(tails)=\frac{1}{2}$$, $$P(\emptyset)=0$$, $$ P(heads, tails)=P(heads)+P(tails)=\frac{1}{2}+\frac{1}{2}=1$$.

Algebra of Probability
The complement of an event $$ A $$ is the event not $$A$$ (that is, the event of $$A$$ not occurring); its probability is given by $$ P(not A) = 1 - P(A)$$. As an example, the chance of not rolling a six on a six-sided die is 1 – (chance of rolling a six)= $$ 1 - \frac{1}{6} = \frac{5}{6}$$.

If both events $$ A $$ and $$ B $$ occur on a single performance of an experiment, this is called the intersection or joint of $$ A $$ and $$ B $$, denoted as $$P(A \cap B)$$. If two events, $$ A $$ and $$ B $$ are independent, then the joint probability is

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$$\displaystyle P(A \mbox{ and }B) = P(A \cap B) = P(A) P(B) $$ (6)
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For example, if two coins are tossed, the chance of both being heads is $$\frac{1}{2}\times\frac{1}{2} = \frac{1}{4}$$.

If either event $$ A $$ or event $$ B $$ or both events occur on a single performance of an experiment this is called the union of the events $$ A $$ and $$ B $$ denoted as $$P(A \cup B)$$.

If two events are mutually exclusive, then the probability of either occurring is

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$$\displaystyle P(A\mbox{ or }B) = P(A \cup B)= P(A) + P(B) $$ (7)
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For example, the chance of rolling a 2 or 3 or 5 on a six-sided die is $$P(\{2,\, 3,\, 5\}) = P(\{2\}) + P(\{3\}) + P(\{5\}) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6}= \frac{1}{2}$$.

If the events are not mutually exclusive then

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$$\displaystyle \mathrm{P}\left(A \hbox{ or } B\right)=\mathrm{P}\left(A\right)+\mathrm{P}\left(B\right)-\mathrm{P}\left(A \mbox{ and } B\right) $$ (8)
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Random variable and vector
Intuitively, Random Variable is used to designate a random outcome (event or sample) in a random experiment, usually denoted in capital letters, e.g., $$\mathbf X $$ is a random variable. It's a numerical description of the outcome of an experiment.

Formally, it is a mapping from a probability space to the real numbers, which is measurable.

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$$\displaystyle \mathbf X:(\Omega ,\mathcal F ) \to (\mathbb R, \mathcal B)$$ $$\displaystyle \omega \mapsto\mathbf X(\omega) $$ (9)
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Where $$(\Omega ,\mathcal F )$$ is event space $$\Omega$$ endowed with $$\sigma$$ - algebra $$\mathcal F$$, $$(\mathbb R, \mathcal B)$$ is set of real numbers $$\mathbb R $$ endowed with " Borel $$\sigma$$ - algebra " $$\mathcal B$$ (sigma-algebra of finite open  subsets of $$\mathbb R$$). (Shao 2007, p.7 .)

$$\mathbf X(\omega) $$ = ( arbitrary ) number selected to represent each event $$\mathbf\omega$$ in $$\mathbf\Omega$$. For Example, typically, in the coin - tossing experiment, we can use number 1 to designate the heads and 0 for the tails, i.e., $$\mathbf X(\rm heads)= 1$$, $$\mathbf X(\rm tails)= 0$$. But it is also possible to select, event though not a good choice, since not as mnemonic as $$\{0,1\}$$, $$\mathbf X(\rm heads)= 5$$, $$\mathbf X(\rm tails)= -3$$.

Example:

$$\mathcal B = \sigma \Big( \{(a,b]: a,b \in \mathbb R \} \Big)$$

$$\{(a,b]: a,b \in \mathbb R \}$$ Set of finite open intervals in $$\mathcal B$$

This choice of $$\mathcal B$$ allows for the probability of

$$\mathbf X \in (a,b]$$, i.e., $$\mathbf P \big(\mathbf X \in (a,b]\big)$$.

(Xiu 2010, p.11 )

In the turbulent flows case, the sample space $$\Omega$$ can be thought of as a set of repeat experiments(samples) to verify, say, a hypothesis or observations on a given flow.

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$$\displaystyle \Omega = \{ \omega_{1},\, \omega_{2},\, ...,\, \omega_{n_{exp}}\} $$
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where $$n_{exp}$$ is the total number of repeated experiments, e.g., until the standard deviation is small enough compared to the mean.

The $$i$$th velocity component(a random variable) at $$(x,\, t)$$ in experiment $$\mathbb\omega_{k}$$ is $$U_{i}(x,\, t,\, \omega_{k})$$.

A random vector $$\displaystyle \mathbf X$$ is composed of real-valued random variables $$\displaystyle X_i, i=1,2,3...$$. A typical n-dimensional random vector can be represented as $$\displaystyle \mathbf X = (X_1, X_2, X_3,...)$$. 【Theorem 1】: Let $$\displaystyle \mathbf X = (X_1,X_2,...,X_n)$$ be a Gaussian random vector with distribution $$\displaystyle N(\mu, \mathbf C)$$ and let $$\displaystyle \mathbf A $$ be an $$\displaystyle m \times n $$ matrix. Then $$\displaystyle \mathbf {AX}^T $$ has an $$\displaystyle N(\mathbf A\mu^T, \mathbf {ACA}^T)$$ distribution.

Fig 2: 2-D random vector

In case events were already representable by real numbers, then the event space is already $$\Omega \equiv \mathbb R$$. It's then not necessary to mention $$\mathbf \omega$$, but directly $$\mathbf X$$. An example of such variable is a velocity component in a turbulent flow.(Pope 2000 )

$$\mathbf P_{\mathbf X}$$ Probability distribution
Probability Distribution is a function that describes the probability of a random variable taking certain values.

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$$\displaystyle P_X = P\circ X^{-1}:\mathcal B \to \mathbb R_{0}^{+} $$ (10)
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Fig 3: Mapping

In practice, only to refer to an open interval in $$\mathcal B$$

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$$\displaystyle {[(a,\, b] \in \mathcal B] \mapsto [P_X((a,\, b]) \in \mathbb R_{0}^{+}]} $$ (11)
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$$\displaystyle P_X((a,b]) \equiv P_X(X( \omega) \in (a,b]) $$ (12) i.e., the probability that $$a < X(\omega) \le b$$.
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$$\mathbf F $$$\mathbf X$ Cumulative distribution function ( CDF )
Cumulative Distribution Function (CDF), or only Distribution Function, describes the probability a real-valued random variable $$\mathbf X$$ with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far" function of the probability distribution.

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$$\displaystyle F_X(x) := P_X((-\infty, x]) = P_X(X \le x) $$ (13)
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For random vectors,
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$$\displaystyle F_X(\mathbf x) := P_X((-\infty, \mathbf x]) = P_X(X_1 \le x_1, X_1 \le x_2,...,X_n \le x_n), \mathbf x = (x_1, x_2,...,x_n) \in \mathbb R^n $$ (14)
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Normal(Gaussian) Distribution

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$$\displaystyle F_X(x) = \frac12\left[\, 1 + \operatorname{erf} \left(\displaystyle \frac{x}{\sqrt{2}} \right) \right]) $$ (15)
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Fig 4: CDF of Normal Distribution

$$f_{\mathbf X}$$ Probability density function ( PDF )
Probability Density Function (PDF), or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the integral of this variable’s density over the region. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.

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$$\displaystyle f_{\mathbf X}(x):=\frac{d}{dx}\mathbf F_{\mathbf X}(x) $$ (24)
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$$\displaystyle P_{\mathbf X} (a < X \leq b) = \int_a^b f_{\mathbf X}(x) \, \mathrm{d}x $$ (25)
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$$\displaystyle \mathbf F_{\mathbf X}(x)=\int_{-\infty}^{x} f_{\mathbf X}(t)dt $$ (26)
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For random vectors,
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$$\displaystyle \mathbf F_{\mathbf X}(\mathbf x)=\mathbf F_{\mathbf X}(x_1,x_2,...,x_n)=\int_{-\infty}^{x_1}...\int_{-\infty}^{x_n} f_{\mathbf X}(t_1,...,t_n)dt_1...dt_n. $$ (27) and
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$$\displaystyle \int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty} f_{\mathbf X}(t_1,...,t_n)dt_1...dt_n = 1 $$ (28) If a vector $$\displaystyle \mathbb X$$ has density $$\displaystyle f_{\mathbb X}$$, then all its subsets have a density, called marginal densities.
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$$\displaystyle f_{X_i}(x_i) = \int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty} f_{\mathbf X}(t_1,...,t_n)dt_1...dt_{i-1}dt_{i+1}...dt_n $$ (29)
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Normal(Gaussian) distribution

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$$\displaystyle f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}}{\rm exp}\left[{-\frac{(x-\mu)^2}{2\sigma^2}}\right] $$ (30)
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Fig 7: Normal(Gaussian) distribution

Binomial distribution

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$$\displaystyle P(X=k)=\binom{n}{k}p^{k}(1-p)^{n-k}, k=0,1,...,n $$ (31)
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Fig 8: Binomial distribution

Poisson distribution
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$$\displaystyle P(X=k)=e^{-\lambda}\frac{\lambda^{k}}{k!}, k=0,1,... $$ (32)
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Fig 9: Poisson distribution

Note:

Notation $$\mathbf X$$ and $$x$$

In recent literature, an uppercase  letter, e.g., $$\mathbf X$$, is used to designate a  random variable, whereas the corresponding  lowercase  letter, e.g., $$x$$, is used to designate the real variable that is the  upper bound  of $$\mathbf X$$.

Kolmogorov, 1933 ; Famous work influencing subsequent mathematical probability and statistics works.

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$$\displaystyle F^{(x)}(a) = \displaystyle \int_{-\infty}^{a}f^{(x)}(a)da $$ (33) (Kolmogorov, 1933 p.24 )
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Expectations and Moments
The expectation, also mean value or the first moment, of a random variable $$\displaystyle X$$ is defined as:

For continuous distribution, with the probability density function $$\displaystyle f_X(x) $$ ,
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$$\displaystyle \mu_X = \mathbb E[X] := \int_{-\infty}^{+\infty}xf_X(x)dx $$ (31) The expectation of $$\displaystyle g(X)$$, where $$\displaystyle g $$ is a real-valued function, is:
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$$\displaystyle \mathbb E[g(X)] := \int_{-\infty}^{+\infty}g(X)f_X(x)dx $$ (32) The $$\displaystyle k$$th moment of random variable $$\displaystyle X$$, where $$\displaystyle k \in \mathbb N$$, is:
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$$\displaystyle \mathbb E[X^k] := \int_{-\infty}^{+\infty}x^kf_X(x)dx $$ (33) The variance of random variable $$\displaystyle X$$, $$\displaystyle \sigma_X^2$$, is:
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$$\displaystyle \sigma_X^2 = var(X) := \int_{-\infty}^{+\infty}(x-\mu_X)^2f_X(x)dx =: \mathbb E[(X-\mu_X)^2] $$ (34) For discrete distribution, with the probability $$\displaystyle p_n=P(X = x_n) $$, the definition of above terms are:
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$$\displaystyle \mu_X = \mathbb E(X) := \sum_{n=1}^{+\infty}x_{n}p_{n} $$ (35)
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$$\displaystyle \mathbb E[g(X)] := \sum_{n=1}^{+\infty}g(X_{n})p_{n} $$ (36)
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$$\displaystyle \mathbb E[X^k] := \sum_{n=1}^{+\infty}x_{n}^{k}p_{n} $$ (37)
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$$\displaystyle \sigma_X^2 = var(X) := \sum_{n=1}^{+\infty}(x_{n}-\mu_X)^2p_{n} =: \mathbb E[(X-\mu_X)^2] $$ (38)
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From equation (34)&(38), we can deduce that
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$$\displaystyle \sigma_X^2 = \mathbb E[(X-\mu_X)^2] = \mathbb E[X^2 - 2X\mu_X + \mu_X^2] = \mathbb E[X^2] - \mathbb E[2X\mu_X] + \mathbb E[\mu_X^2] = \mathbb E[X^2] - \mu_X^2 $$ (39) Variance equals the mean of the square minus the square of the mean.
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The expectation of a random vector $$\displaystyle \mathbf X$$ is
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$$\displaystyle \mu_{\mathbf X}=\mathbb E[\mathbf X] = (\mathbb E[X_1],\mathbb E[X_2],...,\mathbb E[X_n]). $$ (40)
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A very frequently used quantity of random vector is the covariance matrix, which is defined as
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$$\displaystyle \mathbf C_{\mathbf X} = cov(X_i,X_j), i,j = 1,2,...,n. $$ (41) where $$\displaystyle cov(X_i,X_j)=\mathbb E[(X_i-\mu_{X_i})(X_j-\mu_{X_j})]= \mathbb E[X_iX_j]-\mu_{X_i}\mu_{X_j}$$ is the covariance of $$\displaystyle X_i$$ and $$\displaystyle X_j$$. We have $$\displaystyle cov(X_i,X_i) = \sigma_{X_i}^2$$.
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Convergence modes
Given a sequence of random variables $$\displaystyle X_1, X_2,...$$, we defined following convergence modes.

Convergence in distribution, $$\displaystyle X_n \overset{d}{\rightarrow} X$$

For all continuous points $$\displaystyle x$$ of the distribution function $$\displaystyle F_X $$, if we have the relation
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$$\displaystyle F_{X_n}(x) \rightarrow F_X(x), \;\;\; n \rightarrow \infty $$ (42) is satisfied.
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Convergence in distribution is a weak convergence.

Convergence in probability, $$\displaystyle X_n \overset{P}{\rightarrow} X$$

If the probability of the difference between $$\displaystyle X_n $$ and $$\displaystyle X $$, with $$\displaystyle n \rightarrow \infty$$ larger than any positive $$\displaystyle \epsilon $$ tends to zero, then we call $$\displaystyle X_n $$ converges to $$\displaystyle X $$ in probability,which can be written as $$\displaystyle X_n \overset{P}{\rightarrow} X $$.
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$$\displaystyle P(|X_n - X| > \epsilon) \rightarrow 0, \;\;\; n \rightarrow \infty $$ (43) Convergence in probability implies convergence in distribution. The converse is true if and only if $$\displaystyle X = x$$ for some constant $$\displaystyle x$$.
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Almost sure Convergence, $$\displaystyle X_n \overset{a.s.}{\rightarrow} X$$

$$\displaystyle L^p$$ Convergence, $$\displaystyle X_n \overset{L^p}{\rightarrow} X$$

$$\displaystyle $$

title: User:Egm6936.f10/Probability concepts