Please answer 1 to 3 and show all steps in detail.

MPlease upload your answers to Canvas as a single PDF file. 1. Suppose X and Y are two binary random variables. Their joint distribution is given below: Y = 1 Y = 3 Y = 5 X 2 0.07 0.04 0.09 X 3 0.14 0.20 0.07 X = 10 0.13 0.11 0.15 (a) Find the marginal probability mass function of X. (b) Find the marginal probability mass function of Y. (c) Are X and Y independent? (d) What is E [X2 BY]? (e) What are E[Y|X = 2], E[Y|X = 3], and E[Y|X =10]? (f) Demonstrate numerically that E [E [Yl X ]] = E [Y]. ll || 2. Suppose X1, ...,X,._ are Bernoulli random variables with success probability p. n i=1 (a) Find the population expectation of 2:1 GsXs. where a; are constants such that E (b) Suppose X1, ...,X,1 are independent. Find the population variance of 221:1 six... (c) For the rest of this problem, suppose a = 2 and the joint distribtion of X1 and X2 is as follows: X2 = 1] X2 = 1 X1: 30/2 {9/2 11:1 p/2 19/2 Gg=1. i. Find the numerical value of p. i. Find the numerical value of Coo{X1?X2}. iii. Find the numerical value of Var [2X1 X2]. 3. Suppose we have a random sample of Xi m N (are?) for 1' = 1...n. (a) Derive the maximum likelihood estimators of ,u and :72. (b) Is the maximum likelihood estimator of is unbiased? Explain why or why not. (c) Is the maximum likelihood estimator of 02 unbiased? Explain why or why not. 4. Use APPLEDTA for this question. These are phone survey data. where each respondent was asked J .I'H...I.I.I.I\" (. u...l..l...'ll II.I. .Ill!\\