Probability mass function

A probability mass function (PMF) is a function associated with a discrete Random variable that gives the Probability that the variable, when measured, is exactly equal to some value. The PMF describes the Probability distribution that a discrete variable follows.

Given a discrete random variable $X$ , its probability mass function $p_{X}$ is defined as $p_{X}:\Omega_{X}\mapsto[0,1]$ , where $\Omega_{X}$ is $X$ 's sample space. It is equal to

p_{X}(x)=P(X=x)

where $P$ is a measure of probability. By definition of probability, the total probability must be 1:

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

Transformations#

Given a discrete random variable $X$ of Probability mass function $f_{X}$ and an invertible transformation $g(x)$ , we can define the transformed random variable as $Y=g(X)$ . The PMF of $Y$ is

f_{Y}(y)=f_{X}(g^{-1}(y))

The same definition applies to a discrete random vector $\mathbf{X}$ that is invertibly transformed into another random vector $\mathbf{Y}=g(\mathbf{X})$ :

f_{\mathbf{Y}}(\mathbf{y})=f_{\mathbf{X}}(g^{-1}(\mathbf{y}))