# 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 $X\lx@stackrel{{d}}{{=}}Y$.

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