Expected Value of a Discrete Distribution

In the previous lesson, we saw that we could compute a continuous posterior distribution when given a continuous prior and a discrete likelihood function; we hope it is clear how that is useful, but we’d like to switch gears for a moment and look at a different (but also extremely useful) computation: the expected value.

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Computing the Expected Value

We’ll start with a quick refresher on how to compute the expected value of a discrete distribution.

You probably already know what expected value of a discrete distribution is; we’ve seen it before in this series. But in case you don’t recall, the basic idea is: suppose we have a distribution of values of a type where we can meaningfully take an average; the “expected value” is the average value of a set of samples as the number of samples gets very large.

A simple example is: what’s the expected value of rolling a standard, fair six-sided die? You could compute it empirically by rolling $6000d6$ and dividing by $6000$, but that would take a while.

Again, recall that in Dungeons and Dragons, $XdY$ is “roll a fair $Y-sided$ die $X$ times and take the sum”.

We could also compute this without doing any rolling; we’d expect that about $1000$ of those rolls would be $1$, $1000$ ...