Independently and identically distributed

Random variables $X_1,X_2,\dots ,X_n$ are independently and identically distributed (i.i.d.) if they are all Independent and all have the same distribution.

Warning

By distribution, the parameters must be the same as well. Remember when we talk about something like the Normal distribution, we’re really referring to a family of distributions. Therefore, $X\sim \mathcal{N}(1,4)$ and $Y\sim \mathcal{N}(1,9)$ would not be i.i.d., even if they both have the Normal distribution, because ${\sigma }_X\ne {\sigma }_Y$.

• This is equivalent to saying they all have the same CDF. If discrete, they all have the same PMF.
• “Identically distributed” doesn’t mean identical values. I can have two variables $X_1$ and $X_2$ that are i.i.d., but may have different values.