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Let $X$ be a discrete random variable that can take values from 1 to $n$, where $n$ is a large fixed number and also let $p\ll n$ be a fixed number. I am trying to find the expectation $$E[X \mid X \leq p].$$

I know how to compute $E[X \mid X = p]$, but I have no idea how to proceed to compute that expectation.

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Let Y be the random variable given by $$Y = \mathbb{1}_{\{X\leq p\}}. $$

Then the expectation that you are trying to compute is $$\mathbb{E}[X|X\leq p] = \mathbb{E}[X|Y=1].$$

Now you can use the fact that for $W$ and $Z$ discrete random variables we have that $$\mathbb{E}[W|Z=z] = \sum_{w\in \operatorname{Rg}(W)} w\cdot \mathbb{P}(W=w|Z=z)$$ to conclude, by substituting accordingly.

Does this help?

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