Symbols:Abbreviations/P/PMF

Abbreviation: PMF, pmf or p.m.f.

Probability mass function:

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X: \Omega \to \R$ be a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.

Then the probability mass function of $X$ is the (real-valued) function $p_X: \R \to \closedint 0 1$ defined as:

$\quad \forall x \in \R: \map {p_X} x = \begin {cases} \map \Pr {\set {\omega \in \Omega: \map X \omega = x} } & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end {cases}$

where $\Omega_X$ is defined as $\Img X$, the image of $X$.

That is, $\map {p_X} x$ is the probability that the discrete random variable $X$ takes the value $x$.

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): p.m.f.