# Random variables

## CDF examples

### Random variables may be a mix of continuous and discrete

Examples of density functions

# Conditional distributions and expectations

Example: let $X$ and $Y$ be two uniform $[0,1]$ random variables that are independent. Then $$F_{XY}(x,y) = F_X(x)\cdot F_Y(y) = x\cdot y$$

Example: let $X$ and $Y$ be two uniform "logistic" random variables that are independent. Then $$F_{XY}(x,y) = F(x)\cdot F(y) \textrm{ where } F(t) = \frac{1}{1+e^{-t}}$$
In the last example, the joint PDF is $$f_{XY}(x,y) = \frac{d}{dx}F(x)\cdot \frac{d}{dy}F(y) \textrm{ where } \frac{d}{dt}F(t) = \frac{e^{-t}}{(1+e^{-t})^2}$$