Event-Conditioned Random Variables

calendar_today April 21, 2026 article 176 words access_time 1 min replay Modified April 21, 2026

Table of Contents

Conditional Distribution Given an Event

Definition 20.1 (Conditional CDF).

Given event $B$ with $P[B] > 0$, the conditional cumulative distribution function of $X$ is

$$ F_{X\mid B}(x) = P[X \le x \mid B] $$

Definition 20.2 (Conditional PMF).

Given event $B$ with $P[B] > 0$, the conditional probability mass function of $X$ is

$$ P_{X\mid B}(x) = P[X = x \mid B] $$

Definition 20.3 (Conditional PDF).

For a random variable $X$ and an event $B$ with $P[B] > 0$, the conditional PDF of $X$ given $B$ is

$$ f_{X\mid B}(x) = \frac{dF_{X\mid B}(x)}{dx} $$

Theorem 20.4 (Conditional PMF and Conditional PDF from Unconditional).

For a random variable $X$ and an event $B \subset S_X$ with $P[B] > 0$

Discrete

$$ P_{X\mid B}(x) = \begin{cases} \dfrac{P_X(x)}{P[B]} & x \in B \\ 0 & \text{otherwise} \end{cases} $$

Continuous

$$ f_{X\mid B}(x) = \begin{cases} \dfrac{f_X(x)}{P[B]} & x \in B \\ 0 & \text{otherwise} \end{cases} $$

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