**Please upload your answers to Canvas as a single PDF file. 1. Suppose X and Y...
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**Please 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 0.07 0.04 0.09 0.14 0.20 0.07 0.13 0.11 0.15 X = 2 X = 3 X = 10 (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 [X2-3Y]? (e) What are E[Y| X = 2], E [Y| X = 3], and E [Y| X = 10]? (f) Demonstrate numerically that E [E [Y|X]]= E [Y]. 2. Suppose X,..., Xn are Bernoulli random variables with success probability p. (a) Find the population expectation of E-1 aiXi, where a, are constants such that (b) Suppose X,... Xn are independent. Find the population variance of E-1 aiX. (c) For the rest of this problem, suppose n = 2 and the joint distribtion of X and X is as follows: =1 X = 0 X = 1 X=0 p/2 p/2 X=1 p/2 p/2 i. Find the numerical value of p. ii. Find the numerical value of Cov (X1, X). iii. Find the numerical value of Var [2X - X]. 3. Suppose we have a random sample of X ~ N (u,0) for i = 1...n. (2) Derive the maximum likelihood estimators of u and i=1 a = 1. **Please 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 0.07 0.04 0.09 0.14 0.20 0.07 0.13 0.11 0.15 X = 2 X = 3 X = 10 (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 [X2-3Y]? (e) What are E[Y| X = 2], E [Y| X = 3], and E [Y| X = 10]? (f) Demonstrate numerically that E [E [Y|X]]= E [Y]. 2. Suppose X,..., Xn are Bernoulli random variables with success probability p. (a) Find the population expectation of E-1 aiXi, where a, are constants such that (b) Suppose X,... Xn are independent. Find the population variance of E-1 aiX. (c) For the rest of this problem, suppose n = 2 and the joint distribtion of X and X is as follows: =1 X = 0 X = 1 X=0 p/2 p/2 X=1 p/2 p/2 i. Find the numerical value of p. ii. Find the numerical value of Cov (X1, X). iii. Find the numerical value of Var [2X - X]. 3. Suppose we have a random sample of X ~ N (u,0) for i = 1...n. (2) Derive the maximum likelihood estimators of u and i=1 a = 1.
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