Theory

Theory 1

PMF of (any) function of two discrete variables

Suppose $W=g(X,Y)$ and $X,Y$ are discrete RVs.

The PMF of $W$:

$$P_W(w)=\sum _{{g(x,y)=w}^{(x,y)\text{ s.t.}}}{P}_{X,Y}(x,y)$$

CDF of (continuous) function of two continuous variables

Suppose $W=g(X,Y)$ and $X,Y$ are continuous RVs, and $g$ is a continuous function.

The CDF of $W$:

$$F_W(w)=P[W\le w]=\underset{g(x,y)\le w}{\iint }{f}_{X,Y}(x,y)dxdy$$

If desired, one can then compute the PDF of $W$ by differentiating the continuous CDF:

$$f_W(w)=\frac{d}{dw}{F}_W(w)$$