Identically distributed random variables - Imported Knowlpedia Content

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 $A\subseteq\mathbb R$ ,

\mathbb P(X\in A)=\mathbb P(Y\in A).

Remarks

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.