# independent identically distributed

Two random variables $X$ and $Y$ are said to be identically distributed if they are defined on the same probability space $(\Omega,\mathcal{F},P)$, and the distribution function $F_{X}$ of $X$ and the distribution function $F_{Y}$ of $Y$ are the same: $F_{X}=F_{Y}$. When $X$ and $Y$ are identically distributed, we write $X\lx@stackrel{{\scriptstyle d}}{{=}}Y$.

A set of random variables $X_{i}$, $i$ in some index set $I$, is identically distributed if $X_{i}\lx@stackrel{{\scriptstyle d}}{{=}}X_{j}$ for every pair $i,j\in I$.

A collection of random variables $X_{i}$ ($i\in I$) is said to be independent identically distributed, if the $X_{i}$’s are identically distributed, and mutually independent (http://planetmath.org/Independent) (every finite subfamily of $X_{i}$ is independent). This is often abbreviated as iid.

For example, the interarrival times $T_{i}$ of a Poisson process of rate $\lambda$ are independent and each have an exponential distribution with mean $1/\lambda$, so the $T_{i}$ are independent identically distributed random variables.

Many other examples are found in statistics, where individual data points are often assumed to realizations of iid random variables.

Title independent identically distributed IndependentIdenticallyDistributed 2013-03-22 14:27:29 2013-03-22 14:27:29 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 60-00 iid independent and identically distributed identically distributed