Probability mass function

Definition

Let $X$ be a discrete random variable on a probability space $(\Omega, \mathcal{F}, P)$ . The probability mass function of $X$ is the function $p_X \colon \mathbb{R} \to [0, 1]$ with $p_X(x) = P(X = x) \quad \text{for } x \in \mathbb{R}.$ More generally, a probability mass function on a countable set $S$ is a function $p \colon S \to [0, 1]$ with $\sum_{s \in S} p(s) = 1$ .

Remarks

The set $\{x : p_X(x) > 0\}$ is countable, and $\sum_x p_X(x) = 1$ . For each Borel set $B$ , $P(X \in B) = \sum_{x \in B} p_X(x)$ . For a fair die, $p_X(k) = 1/6$ for $k = 1, \dots, 6$ and $p_X(x) = 0$ for other $x$ .

Tag	`0WF4EJP91W`
URI	`m2://0WF4EJP91W`
Kind	definition
Writing	W0 stub
MSC	60E05
Defines	Probability mass function
Uses	Discrete random variable, Random variable

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0WF4EJP91W

URI

m2://0WF4EJP91W

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definition

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W0 stub

MSC

60E05

Defines

Probability mass function

Uses

Discrete random variable, Random variable

Probability mass function

Definition

Remarks

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