Grinstead-Snell-Probability · Problem 6.3.17

Sums of Random Variables

Mathematics · covariance

Problem

Let XX and YY be random variables. The covariance Cov(X,Y)\mathrm{Cov}(X, Y) is defined by (see Exercise 6.2.23) cov(X,Y)=E((X−μ(X))(Y−μ(Y))) .\mathrm{cov}(X, Y) = E((X - \mu(X))(Y - \mu(Y)))\,. (a) Show that cov(X,Y)=E(XY)−E(X)E(Y)\mathrm{cov}(X, Y) = E(XY) - E(X)E(Y). (b) Using (a), show that cov(X,Y)=0\mathrm{cov}(X, Y) = 0, if XX and YY are independent. (Caution: the converse is not always true.) (c) Show that V(X+Y)=V(X)+V(Y)+2cov(X,Y)V(X + Y) = V(X) + V(Y) + 2\mathrm{cov}(X, Y).

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