So that these posts don’t get too long, I’ll split them up when there’s more theorems/examples/etc. I want to add. The first new topic is the “law of the unconscious statistician”:

**Theorem 1** * If is a random variable with and is a Borel-measurable function such that , then *

*
** where is the distribution function of . *

*Proof:* Define , that is, for each , . Then the distribution of is

where

Note that is a Borel-measurable set because is a Borel-measurable function. For all such sets , define

Then is a probability measure on , and

It follows that

When , we have the familiar formula

When takes countably many values (is discrete), this reduces to

where is the set of values that takes. When is absolutely continuous (i.e. is a continuous random variable that admits a density), this reduces to

where is the density of .

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