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

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

## Mathematics Subject Classification

60-00*no label found*

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