Identically distributed random variables

A pair of identically distributed random variables $X$ and $Y$ is a pair of random variables whose distribution laws agree; equivalently, for every Borel set $A\subseteq\mathbb R$,

Identical distribution compares only marginal behavior and does not impose independence . When the relevant moments exist, identically distributed variables have the same expectation and the same variance .

Examples:

• If $X$ is the indicator of “heads” on the first fair coin toss and $Y$ is the indicator of “heads” on the second toss, then $X$ and $Y$ are identically distributed (both are Bernoulli$(1/2)$).
• If $U$ is uniform on $[0,1]$ and $Y=1-U$ on the same probability space, then $U$ and $Y$ are identically distributed even though they are completely dependent.