Proofs of Selected Theorems

A.1 A Proof of Theorem 3.7.5

Let X and Y be continuous or discrete random variables. Define the joint cdf of X and Y to be

FXY(x, y) = P(Xx, Yy)

and the marginal cdfs to be

FX(x) = P(Xx) and FY(y) = P(Yy).

Then X and Y are independent if and only if

FXY(x, y) = FX(x) · FY(y).

We use this idea to prove Theorem 3.7.5.

Theorem A.1.1 Let X and Y be independent discrete or continuous random variables, and let g(x) and h(y) be any functions. Then the random variables Z = g(X) and W = h(Y) are independent.

Proof. Let z and w be any elements in the ranges of Z and W, respectively. Also let RX and RY denote the ranges of X and Y, respectively. Define the sets

Az = {xRX : g(x) ≤ z} and Bw = {y

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