**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*denote the ranges of

_{Y}*X*and

*Y,*respectively. Define the sets

*A _{z}* = {

*x*∈

*R*:

_{X}*g*(

*x*) ≤

*z*} and

*B*= {

_{w}*y*∈

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