Independently and Identically distributed random variables

Question

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

Answer 1

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. $$F_{XY}(x,y)=F_X(x)F_Y(y).$$