Question

Suppose we have a random variable X such that X-1 with probability 1/2 and X =-1 with probability 1/2·We also have another random variable Y such that Y-X with probability 3/4 and YX with probability 1/4. What is the covariance between them, Cov(X, Y)?

Answer 1

$Cov(X,Y)=E(XY)-E(X)E(Y).....(1)\\ E(XY)=E(X^2)\times \frac{3}{4}-E(X^2)\times \frac{1}{4}=\frac{1}{2}E(X^2)=\frac{1}{2}\left(1\times \frac{1}{2}-1\times \frac{1}{2} \right )=0\\ E(X)=1\times \frac{1}{2}-1\times \frac{1}{2}=0\\ Hence~Cov(X,Y)=0$