Question

If X is a random variable and ‘a’ is any constant, then prove that E(ax) = aE(X) and var(ax) = a2var(x).

Answer 1

Proof : (i) E(ax) = Σax(px)

By definition E(X) = ΣxP(X) = aΣxP(X)

∴ E(ax) = aE(x)

(ii) var(ax) = E[ax – aE(x)]2

by definition of var (x) = E [x – E(x)]2 = E [ax – aE(x)]2 = a2E [x – E(x)]2

var(ax)=a2 var(x).