Question

7. Let X, and X, be indicators of independent events with probabilities 1/2 and 1/3. respectively. a) Display the joint distribution table of X2 + Xs and X - Xs. b) Calculate E(X3 - X) C) Are X and Xs uncorrelated? Prove your answer.

Answer 1

Suppose A and B are independent events then X2(w) = 1 if we A= P(WE A) = 1/2 = 0 otherwise X3(w) = 1 if we BP(W E B) = 1/3 = 0 otherwise P(X2 = 1, X3 = 1) = P(ANB) = P(A)P(B) P(X2 = 1, X3 = 0) = P(ANB)= P(A)P(B9) P(X2 = 0, X3 = 1) = P(An B) = P(A)P(B) P(X2 = 0, X3 = 0) = P(An B) = P(A)P(B) Hence X2 and X3 are independent.

7 a.

$Let~Z_1=X_2+X_3~and~Z_2=X_2-X_3.~The~joint~pdf~of~(Z_1,Z_2)~is\\\\ P(Z_1=0,Z_2=0)=P(X_2=0,X_3=0)\\\\ =P(X_2=0)P(X_3=0)=\frac{2}{6}~(Since~X_2~and~X_3~are~independent)\\\\ P(Z_1=1,Z_2=-1)=P(X_2=0,X_3=1)\\\\ =P(X_2=0)P(X_3=1)=\frac{1}{6}~(Since~X_2~and~X_3~are~independent)\\\\ P(Z_1=1,Z_2=1)=P(X_2=1,X_3=0)\\\\ =P(X_2=1)P(X_3=0)=\frac{2}{6}~(Since~X_2~and~X_3~are~independent)\\\\ P(Z_1=2,Z_2=0)=P(X_2=1,X_3=1)\\\\ =P(X_2=1)P(X_3=1)=\frac{1}{6}~(Since~X_2~and~X_3~are~independent)\\\\ P(Z_1=z_1,Z_2=z_2)=0~otherwise.$