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 I also mean the parameters must be the same. 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.