C3 - conditional expectation

conditional PMF

let $X$ be a discrete RV with $p_{X} (\cdot)$ defined over a countable sample space, and $A$ be an event such that $P (A) > 0$ , then the conditional PMF of $X$ given event $A$ is defined as:

p_{X | A} = P (X = x | A) = \frac{P ((X = x) | A)}{P (A)}

\sum_{x} p_{X | A} (x) = \sum_{x \in A} p_{X | A} (x) = 1

conditional expectation

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