Consider four multi-valued random variables C (campus), G (grade), M (major), and Y (year). We know...
Consider four multi-valued random variables C (campus), G (grade), M (major), and Y (year). We know that none of these variables are independent. We are provided the probability tables for the following joint, marginal, and conditional probabilities.
P(Y) P(M) P(G,Y) P(C|Y) P(C,M) P(Y|M)
For example, we are told:
P(M=compSci) = 0.3, P(M=psych) = 0.2, P(M=bio)=0.2,
P(M=business)=0.1P(G=A,Y=freshman)=0.03, P(G=B,Y=freshman)=0.12 ,
... P(G=F , Y=senior)=0.08[corrected Jan 18, 11am]
We are not provided any other probability tables; for example, we are not given values for: P(G=B) or P(M=psych, Y=junior)
Explain how to combine probabilities from above to compute each
probability below, or write“not possible” if it is not
possible.
For example: P(Y) = ∑? ?(?, ? = ?)
a) P(M=compSci, Y=sophomore)
b) P(G=B | C=LC, M=business)
c) P(C=RH | Y=freshman)
d) P(G=B, M=bio)
Homework Answers
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Consider four multi-valued random variables C (campus), G
(grade), M (major), and Y (year). We know...
Consider four multi-valued random variables C (campus), G (grade), M (major), and Y (year). We know...
Consider four multi-valued random variables C (campus), G (grade), M (major), and Y (year). We know that none of these variables are independent. We are provided the probability tables for the following joint, marginal, and conditional probabilities. P(Y) P(M) P(G,Y) P(C|Y) P(C,M) P(Y|M) For example, we are told: P(M=compSci) = 0.3, P(M=psych) = 0.2, P(M=bio)=0.2, P(M=business)=0.1P(G=A,Y=freshman)=0.03, P(G=B,Y=freshman)=0.12 , ... P(G=F , Y=senior)=0.08[corrected Jan 18, 11am] We are not provided any other probability tables; for example, we are not given values...
. Suppose we have the following joint distribution for random variables X and Y 2 0.1...
. Suppose we have the following joint distribution for random variables X and Y 2 0.1 0.2 0.1 4 0 0.3 0.1 6 0 0 0.2 (a) Find p(X). That is find the marginal distribution of X. (b) Find p(Y). That is find the marginal distribution of Y (c) Find the distribution of X conditional on Y = 3. (d) Find the distribution of X conditional on Y 2 (e) Are X and Y independent? You should be able to...
Suppose that we have two independent binomial random variables X ~Binomial(n, px) and Y ~ Binomial(m,Pv)....
Suppose that we have two independent binomial random variables X ~Binomial(n, px) and Y ~ Binomial(m,Pv). You can assume that the MLE's are -X/n and p,-Y/m. (a) Find the MLE for p under the assumption that p (b) Find the LRT statistic T for testing p,-py HA:p.Ру vs. (c) Evaluate the value of this statistic if n 353, X 95, m -432, and Y 123. (d) Compare the answer from part (c) to a critical value from a x2 with...
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Consider IS-LM Model: Real Sector: Y=C+I+G C=a+b (1-t) Y I=d-ei G=GO t-income tax rate i- rate...
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and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P...
and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P(X>x) P(x (3 = Similarly, for all y [0,1,2,...^, show that Show your working only for one of the two identities that are pre- sented above. Hint: You may use the following identity without proving it. For any non-negative integer (, we have:...
part C (b) Consider the experiment on pp. 149-156 of the online notes tossing a coin...
part C
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Assume that we have three independent observations: where Xi ~ Binomial(n 7,p) for i E { 1.2.3). The value of p E (0, 1) is not known. When we have observations like this from different, independent...
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Problem 2. Consider the following joint probabilities for the two variables X and Y. 1 2...
Problem 2. Consider the following joint probabilities for the two variables X and Y. 1 2 3 .14 .25 .01 2 33 .10 .07 3 .03 .05 .02 Find the marginal probability distribution of Y and graph it. Show your calculations. b. Find the conditional probability distribution of Y (given that X = 2) and graph it. Show your calculations. c. Do your results in (a) and (b) satisfy the probability distribution requirements? Explain clearly. d. Find the correlation coefficient...
Need help with stats true or false questions Decide (with short explanations) whether the following statements are true or false a) We consider the model y-Ao +A(z) +E. Let (-0.01, 1.5) be a 95% con...
Need help with stats true or false questions
Decide (with short explanations) whether the following statements are true or false a) We consider the model y-Ao +A(z) +E. Let (-0.01, 1.5) be a 95% confidence interval for A In this case, a t-test with significance level 1% rejects the null hypothesis Ho : A-0 against a two sided alternative. b) Complicated models with a lot of parameters are better for prediction then simple models with just a few parameters c)...
. Suppose we have the following joint distribution for random variables X and Y 2 0.1 0.2 0.1 4 0 0.3 0.1 6 0 0 0.2 (a) Find p(X). That is find the marginal distribution of X. (b) Find p(Y). That is find the marginal distribution of Y (c) Find the distribution of X conditional on Y = 3. (d) Find the distribution of X conditional on Y 2 (e) Are X and Y independent? You should be able to...
Suppose that we have two independent binomial random variables X ~Binomial(n, px) and Y ~ Binomial(m,Pv). You can assume that the MLE's are -X/n and p,-Y/m. (a) Find the MLE for p under the assumption that p (b) Find the LRT statistic T for testing p,-py HA:p.Ру vs. (c) Evaluate the value of this statistic if n 353, X 95, m -432, and Y 123. (d) Compare the answer from part (c) to a critical value from a x2 with...
4. Points = 18. Consider IS-LM Model: Real Sector: Y=C+I+G C = a +b (1-t) Y I=d-ei G=Go t-income tax rate i-rate of interest Money Market: Ma=M Ma= kY-li Mg = Mo Mo - exogenous stock of money 1) Setup the system of solutions in general form, with variables vector in the following order: Y, C, I, i; (6 points) 2) Now, suppose we have the following values of parameters: a = 10; b = 0.7; t = 0.2; d...
Consider IS-LM Model: Real Sector: Y=C+I+G C=a+b (1-t) Y I=d-ei G=GO t-income tax rate i- rate of interest Money Market: Ma=M Ma=ky-li M = Mo Mo - exogenous stock of money 1) Setup the system of solutions in general form, with variables vector in the following order: Y, C, I, i; (6 points) 2) Now, suppose we have the following values of parameters: a= 10; b = 0.7; t = 0.2; d = 25; k = 0.25; 1 = 0.04;...
and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P(X>x) P(x (3 = Similarly, for all y [0,1,2,...^, show that Show your working only for one of the two identities that are pre- sented above. Hint: You may use the following identity without proving it. For any non-negative integer (, we have:...
part C
(b) Consider the experiment on pp. 149-156 of the online notes tossing a coin three times). Consider the following discrete random variable: Y = 2[number of H-3[number of T). (For example, Y (HHT) = 2.2-3.1=1, while Y (TTH) = 2.1-3.2 = -4.) Repeat the analysis found on pp. 149-156. That is, (i) find the range of values of Y: (ii) find the value of Y(s) for each s ES: (iii) find the outcomes in the events A -Y...
Assume that we have three independent observations: where Xi ~ Binomial(n 7,p) for i E { 1.2.3). The value of p E (0, 1) is not known. When we have observations like this from different, independent ran- dom variables, we can find joint probabilities by multiplying together th ndividual probabilities. For example This should remind you the discussion on statistical independence of random variables that can be found in the course book (see page 22) Answer the following questions a...
Problem 2. Consider the following joint probabilities for the two variables X and Y. 1 2 3 .14 .25 .01 2 33 .10 .07 3 .03 .05 .02 Find the marginal probability distribution of Y and graph it. Show your calculations. b. Find the conditional probability distribution of Y (given that X = 2) and graph it. Show your calculations. c. Do your results in (a) and (b) satisfy the probability distribution requirements? Explain clearly. d. Find the correlation coefficient...
Need help with stats true or false questions
Decide (with short explanations) whether the following statements are true or false a) We consider the model y-Ao +A(z) +E. Let (-0.01, 1.5) be a 95% confidence interval for A In this case, a t-test with significance level 1% rejects the null hypothesis Ho : A-0 against a two sided alternative. b) Complicated models with a lot of parameters are better for prediction then simple models with just a few parameters c)...
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