Essentials of Mathematical Statistics

APPENDIX A

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

F_XY(x, y) = P(X ≤ x, Y ≤ y)

and the marginal cdfs to be

F_X(x) = P(X ≤ x) and F_Y(y) = P(Y ≤ y).

Then X and Y are independent if and only if

F_XY(x, y) = F_X(x) · F_Y(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 R_X and R_Y denote the ranges of X and Y, respectively. Define the sets

A_z = {x ∈ R_X : g(x) ≤ z} and B_w = {y ∈