Question

Let X and Y be independent normal random variables with parameters E[X] =ux, E[Y] = uy and Var(X) = x, Var(Y) = Oy. Indicate whether each of the following statements is true or false. Notation: fx,y (x, y), fx(x), fy (v) denote the joint and marginal PDFs of X and Y , respectively; $(x) is the CDF of a standard normal random variable with zero mean and unit variance. E[XY]=0

Answer 1

Solution:

If we know that if X and Y are independent events then the Covariance between X and Y becomes zero and we defined covariance between X and Y as

Cov(X,Y) =E(XY)-E(X)E(Y)

Cov(X,Y)=0

E(XY)-E(X)E(Y)=0

E(XY)=E(X)E(Y) this condition hold if X and Y are independent events.

so above statement is true that E(XY)=0 if X and Y are independent events.

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