Question

6) 112 pts] Let X and U denote random variables, and suppose that E[uXCERand that E[X]-aER. (a) What is E[U]? Provide a proof of your answer (b) What is EjXu]? Provide a proof of your answer. (c) What is Cov X,Uj? Provide a proof of your answer. (d) Is U mean independent of X? Explain briefly. (e) Is U uncorrelated with X? Explain briefly. () Do you have enough information to determine whether X is independent of u? Ex plain briefly

Answer 1

$6.~a.~E(U)=EE(U|X)=E(c)=c\\ b.~E(XU)=E(XE(U|X))=E(cX)=cE(X)=ac\\ c.~Cov(X,U)=E(XU)-E(U)E(X)=ac-ac=0\\$

$d.~U~and~X~may~not~be~independent.~ Consider following~Example:\\ Joint~distribution~of~(X,U)~is:\\ \begin{matrix} &U=0 &U=1 &U=2 \\ X=0& 0.1 & 0.1 &0.1 \\ X=1& 0.1 &0.2 &0.1 \\ X=2& 0.1 & 0.1 &0.1 \end{matrix}\\ P(X=0)=0.1+0.1+0.1=0.3~and~P(U=u|X=x)=\frac{P(U=u,X=x)}{P(X=x)}\\ P(U=0|X=0)=1/3,~P(U=1|X=0)=1/3,~P(U=2|X=0)=1/3\\ E(U|X=0)=1/3+2/3=1\\ P(X=1)=0.1+0.2+0.1=0.4\\$

$P(U=0|X=1)=1/4,~P(U=1|X=0)=1/2,~P(U=2|X=0)=1/4\\ E(U|X=1)=1/2+2/4=1\\$

$Similarly,~P(X=2)=0.3\\ P(U=0|X=2)=1/3,~P(U=1|X=2)=1/3,~P(U=2|X=2)=1/3\\ E(U|X=2)=1/3+2/3=1\\ Hence~E(U|X)=constant\\ but~P(U=0,X=0)=0.1\neq P(U=0)P(X=0)=0.3*0.3\\ Hence~U~and~X~are~not~independent.$

$(e)~Since~Cov(U,X)=0~hence~U~and~X~are~uncorrelated.\\ (f)~No, ~the~information~is~not~enough.~See~example~given~in~Part~(d).$