Final Exams Show All Your Work Calculators Are Allowed

Final Examshow All Your Work Calculators Are Allowedafter You Finis

Final Examshow All Your Work Calculators Are Allowedafter You Finis

FINAL EXAM Show all your work. Calculators are allowed. After you finish, scan and upload your exam as a pdf file. (1) (a) (5pt) Suppose that there are 10 red balls and 12 blue balls inside a box. Draw five balls at random without replacement. What is the probability that the second and the third balls are red, but all the other balls are blue? (b) (5pt) Consider a factory which produces electronic components.

Assume that, over a long run, 3% of the components are faulty, and that component faults are independent of one another. If this factory produces a batch of 713 components, which is the most likely number of faulty components in this batch? (2) (a) (5pt) Let f(x) be the density function for a continuous random variable X. Suppose that f(x) = ( c cos x if x 2 (⇡/2, ⇡/2) 0 otherwise for some constant c to be determined. Compute the constant c. Also, find P(⇡/2  X  ⇡/3). (b) (5pt) Suppose that X and Y are bivariate standard normal variables with correlation 2/3.

Find an expression for P(2X +Y  5) in terms of the cumulative distribution function of standard normal distribution. (3) (a) (4pt) Suppose that X is a random variable with distribution P(X = 1/4) = 2/3, P(X = 3/4) = 1/3. Suppose that, given X = x, the random variable Y is binomial(3, x)-distributed. Compute the conditional distribution of X given Y = 2. (b) (4pt) Let X and Y have joint density f(x, y) = ( 2x + 2y 4xy if 0

Use the above equation to show that the density fX+Y of X + Y is fX+Y (t) = Z 1 1 fX(x)fY (t x) dx = Z 1 1 fX(t y)fY (y) dy. (b) (4pt) Suppose that X and Y are independent uniform(0, 2) random variables. Use the result in part (a), or otherwise (such as techniques in Section 5.1), compute the density of X + Y . (c) (2pt) Find an example of jointly continuous random variables U and V such that the marginal den- sities of U and V are both uniform(0, 1) distribution, but U + V must have a different density as in part (b). 1 Problem Set ) In this problem, leave your answers in terms of the cumulative distribution function of a standard normal random variable when needed. Let X be a standard normal random variable.

Let W be a discrete random variable independent of X. Assume that P(W = 1) = P(W = 1) = 12 . Let Y = WX. (a) (4pt) Let x 2 R. Consider the indicator Y x of the event {Y  x}. Compute the conditional expectation E[ Y x|W ]. (b) (3pt) Use part (a) to show that Y has standard normal distribution. (c) (4pt) Compute the covariance Cov(X, Y ).

Explain why X and Y do not have a bivariate normal distribution. 2 Wk 2 Discussion - Self-Disclosure Respond to classmate’s posts must be a minimum of 175 words: D.B. Is self-disclosure an important component? Yes, but it has it’s time and place! Essentially, self-disclosure bonds the relationship between client and counselor.

Along with not only unifying such relationship, it enables rapport to be built as well. If you think about it, trust is an important component in many individuals’ lives. This is something to be considerate of. If a person’s trust has been broken numerous times, they are less likely to be trustworthy of people they come in contact with. Another form of “self-disclosure†that comes to mind happens in the informed consent portion of treatment.

Informed consent involves providing honesty to guidelines one must abide by. I view this as a form of self-disclosure. Even though informed consent is required, I think this portion allows clients to see honesty given by their counselor and doesn’t leave them blindsided to the “can’s and cant’s†of treatment. Going back to my response about self-disclosure having its time and place, is also relatable to why it’s essential to be mindful of such disclosure in the first place. We connect with people through our similarities.

I believe this can be an effective tool used in the professional approach. With the stigma of mental illness, being vulnerable can be hard. However, sometimes it’s beneficial to know others (especially those you seek counsel from) also have suffered from relevant issues to yours, creating a likeness between the two. This is when I believe it is okay to disclose information, when it’s done mindfully to benefit the other person. Self-disclosure becomes harmful when a professionals discloses too much.

Resulting in the client listening while the counselor does all the talking. Other forms of harmful disclosure involve: feelings of likeness between client and counselor spiral to inappropriate behaviors. Body language is huge, and sadly can be misused as a nonverbal form of self-disclosure. The actions of this may lead a professional to dual-relationships, intimate relationships, or causing harm to their client. Truthfully, disclosure is a complex topic.

Being mindful of when and when not to share personal information is a tool needing practice each and every day. Instructor The dangers of self disclosure are accurately summarized in your post. I want to note that I suggest that beginning counselors avoid self disclosure until experience is gained and a understanding of when and what type of self disclosure is appropriate. I have seen sessions with student counselors where a small insert of self disclosure took the session completely away from the client and then I could see the student counselor trying to 'fix' the area that had been self disclosed in the life the client - and we definitely want to avoid that. Instructor The client’s problem: Joe has been married for eight years.

Mary, his spouse, was his high school sweetheart. They married right after high school and had their first child. Four years ago, they had a second child. Both Joe and Mary believe they made a mistake in getting married so early. But they do still love each other.

Lately they have found themselves talking about what it would have been like if they had dated other people before getting married. They have even considered a marital separation to test the strength of their love. Develop an outline of questions that would assist you in identifying the following: A. Client feelings B. Client thoughts C. Client behaviors D. Client physical/somatic complaints E. Client interpersonal aspects

Final Examshow All Your Work Calculators Are Allowedafter You Finis

Final Examshow All Your Work Calculators Are Allowedafter You Finis

FINAL EXAM Show all your work. Calculators are allowed. After you finish, scan and upload your exam as a pdf file. (1) (a) (5pt) Suppose that there are 10 red balls and 12 blue balls inside a box. Draw five balls at random without replacement. What is the probability that the second and the third balls are red, but all the other balls are blue? (b) (5pt) Consider a factory which produces electronic components.

Assume that, over a long run, 3% of the components are faulty, and that component faults are independent of one another. If this factory produces a batch of 713 components, which is the most likely number of faulty components in this batch? (2) (a) (5pt) Let f(x) be the density function for a continuous random variable X. Suppose that f(x) = ( c cos x if x 2 (⇡/2, ⇡/2) 0 otherwise for some constant c to be determined. Compute the constant c. Also, find P(⇡/2  X  ⇡/3). (b) (5pt) Suppose that X and Y are bivariate standard normal variables with correlation 2/3.

Find an expression for P(2X +Y  5) in terms of the cumulative distribution function of standard normal distribution. (3) (a) (4pt) Suppose that X is a random variable with distribution P(X = 1/4) = 2/3, P(X = 3/4) = 1/3. Suppose that, given X = x, the random variable Y is binomial(3, x)-distributed. Compute the conditional distribution of X given Y = 2. (b) (4pt) Let X and Y have joint density f(x, y) = ( 2x + 2y 4xy if 0

Use the above equation to show that the density fX+Y of X + Y is fX+Y (t) = Z 1 1 fX(x)fY (t x) dx = Z 1 1 fX(t y)fY (y) dy. (b) (4pt) Suppose that X and Y are independent uniform(0, 2) random variables. Use the result in part (a), or otherwise (such as techniques in Section 5.1), compute the density of X + Y . (c) (2pt) Find an example of jointly continuous random variables U and V such that the marginal densities of U and V are both uniform(0, 1) distribution, but U + V must have a different density as in part (b). 1 Problem Set ) In this problem, leave your answers in terms of the cumulative distribution function of a standard normal random variable when needed. Let X be a standard normal random variable.

Let W be a discrete random variable independent of X. Assume that P(W = 1) = P(W = 1) = 1/2. Let Y = WX. (a) (4pt) Let x 2 R. Consider the indicator Y x of the event {Y  x}. Compute the conditional expectation E[ Y x|W ]. (b) (3pt) Use part (a) to show that Y has standard normal distribution. (c) (4pt) Compute the covariance Cov(X, Y ).

Explain why X and Y do not have a bivariate normal distribution. 2 Wk 2 Discussion - Self-Disclosure Respond to classmate’s posts must be a minimum of 175 words: D.B. Is self-disclosure an important component? Yes, but it has it’s time and place! Essentially, self-disclosure bonds the relationship between client and counselor.

Along with not only unifying such relationship, it enables rapport to be built as well. If you think about it, trust is an important component in many individuals’ lives. This is something to be considerate of. If a person’s trust has been broken numerous times, they are less likely to be trustworthy of people they come in contact with. Another form of “self-disclosure†that comes to mind happens in the informed consent portion of treatment.

Informed consent involves providing honesty to guidelines one must abide by. I view this as a form of self-disclosure. Even though informed consent is required, I think this portion allows clients to see honesty given by their counselor and doesn’t leave them blindsided to the “can’s and cant’s†of treatment. Going back to my response about self-disclosure having its time and place, is also relatable to why it’s essential to be mindful of such disclosure in the first place. We connect with people through our similarities.

I believe this can be an effective tool used in the professional approach. With the stigma of mental illness, being vulnerable can be hard. However, sometimes it’s beneficial to know others (especially those you seek counsel from) also have suffered from relevant issues to yours, creating a likeness between the two. This is when I believe it is okay to disclose information, when it’s done mindfully to benefit the other person. Self-disclosure becomes harmful when a professionals discloses too much.

Resulting in the client listening while the counselor does all the talking. Other forms of harmful disclosure involve: feelings of likeness between client and counselor spiral to inappropriate behaviors. Body language is huge, and sadly can be misused as a nonverbal form of self-disclosure. The actions of this may lead a professional to dual-relationships, intimate relationships, or causing harm to their client. Truthfully, disclosure is a complex topic.

Being mindful of when and when not to share personal information is a tool needing practice each and every day. Instructor The dangers of self disclosure are accurately summarized in your post. I want to note that I suggest that beginning counselors avoid self disclosure until experience is gained and a understanding of when and what type of self disclosure is appropriate. I have seen sessions with student counselors where a small insert of self disclosure took the session completely away from the client and then I could see the student counselor trying to 'fix' the area that had been self disclosed in the life the client - and we definitely want to avoid that. Instructor The client’s problem: Joe has been married for eight years.

Mary, his spouse, was his high school sweetheart. They married right after high school and had their first child. Four years ago, they had a second child. Both Joe and Mary believe they made a mistake in getting married so early. But they do still love each other.

Lately they have found themselves talking about what it would have been like if they had dated other people before getting married. They have even considered a marital separation to test the strength of their love. Develop an outline of questions that would assist you in identifying the following: A. Client feelings B. Client thoughts C. Client behaviors D. Client physical/somatic complaints E. Client interpersonal aspects

Final Examshow All Your Work Calculators Are Allowedafter You Finis

Final Examshow All Your Work Calculators Are Allowedafter You Finis

FINAL EXAM Show all your work. Calculators are allowed. After you finish, scan and upload your exam as a pdf file.

From the original included instructions, the core assignment questions are as follows:

1. Suppose that there are 10 red balls and 12 blue balls inside a box. Draw five balls at random without replacement. What is the probability that the second and the third balls are red, but all the other balls are blue?
2. Consider a factory which produces electronic components. Assume that, over a long run, 3% of the components are faulty, and component faults are independent. If the factory produces 713 components, what is the most likely number of faulty components?
3. Let f(x) be the density function for a continuous random variable X, where f(x) = c cos x for x in the interval (−π/2, π/2), and zero otherwise. Compute the constant c and find P(−π/2 ≤ X ≤ −π/3).
4. Suppose X and Y are bivariate standard normal variables with correlation 2/3. Find an expression for P(2X + Y ≥ 5) in terms of the standard normal cumulative distribution function.
5. Suppose X takes values P(X=1/4)=2/3 and P(X=3/4)=1/3; given X=x, Y has a binomial distribution with parameters n=3, p=x. Compute the conditional distribution of X given Y=2.
6. Let X and Y be joint density f(x, y) = 2x + 2y for 0
7. Suppose X and Y are independent uniform(0, 2) random variables. Using the derived formulas for sums of independent variables, compute the density of X+Y.
8. Let X be a standard normal variable; W be a discrete variable with P(W=1)=1/2, and Y=WX. Compute the conditional expectation E[ Y ≤ x | W ], demonstrate that Y has the same distribution as X, compute Cov(X,Y), and explain why (X, Y) are not jointly normal.
9. Discuss the importance of self-disclosure in clinical settings, its appropriate usage, and potential risks based on a shared classmate’s post.
10. Given a client, Joe, married for 8 years and considering separation, develop questions to identify feelings, thoughts, behaviors, physical complaints, and interpersonal aspects.

Paper For Above instruction

The following paper systematically addresses the various statistical and psychological questions outlined above. It begins with probability computations involving drawing balls without replacement, then proceeds to analyze fault probability in manufacturing, and finally explores concepts in continuous and discrete distributions, as well as the implications of self-disclosure in clinical practice. This comprehensive response elucidates each question with detailed explanations and relevant formulas or reasoning.

Probability of Drawing Specific Balls

In the first scenario, there are 10 red balls and 12 blue balls, totalling 22 balls. Drawing five balls without replacement involves calculating the probability that the second and third are red, while the other three are blue. The probability that the first ball