Given that "X" and "Y"are two independent random variables
"E(X)=1, E(Y)=0, Var(X)=4, Var(Y)=2""W=2X+Y+1, Z=3X+Y"a) Distributive property of expected value: for any rv "X,Y" and any constants,"a,b,c"
"E(W)=E(2X+Y+1)=2E(X)+E(Y)+1=""=2(1)+0+1=3"
"E(Z)=E(3X+Y)=3E(X)+E(Y)=""=3(1)+0=3"
b) If "X" is any random variable and "c" is any constant, then
If "X"and "Y"are two independent random variables, then
c) For any two random variables "X" and "Y,"
If "X" and "Y" are independent, then "Cov(X,Y)=0." by observing that "E(XY)=E(X)E(Y)"
Distributive property of covariance: for any rv "X,Y,Z" and any constants, a, b, c,
"Cov(W,Z)=Cov(Z,W)=""=Cov(2X+Y+1,3X+Y)=""=2Cov(X,3X+Y)+Cov(Y,3X+Y)=""=6Cov(X,X)+2Cov(X,Y)+3Cov(Y,X)+Cov(Y,Y)=""=6Var(X)+5Cov(X,Y)+Var(Y)"
"Cov(W,Z)=6(4)+5(0)+2=26"
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