# Independently and Identically distributed random variables

What does it mean for two random variables to be "identically distributed"? Does it mean that they come from the same probability distribution?

## 1 Answer

Suppose $$X$$ and $$Y$$ are iid random variables. The 'identically distributed' part means both random variables have the same distribution function (cdf). Formally this can be stated as $$F_X(z)=F_Y(z),$$ where $$F_X(\cdot)$$ and $$F_Y(\cdot)$$ are the marginal cdfs of $$X$$ and $$Y$$, respectively. The 'independently distributed' part means the joint cdf of $$X$$ and $$Y$$, denoted $$F_{XY}(x,y)$$ is simply the product of the marginal cdfs $$F_X(x)$$ and $$F_Y(y)$$, i.e. $$F_{XY}(x,y)=F_X(x)F_Y(y).$$