Probability Mass Function

probability-theory

Definition

Probability Mass Function

Let $X:\Omega \to \mathcal{X}$ be a discrete random variable on a probability space $(\Omega ,\mathfrak{S},P)$, where $\mathcal{X}$ is finite or countable. Its probability mass function is

$$p_X:\mathcal{X}\to [0,1],p_X(x)=P(X=x)=P(X^{-1}(\{x\})).$$

It assigns probability mass to each possible value of $X$ and satisfies

1. Non-negativity: $p_X(x)\ge 0$ for every $x\in \mathcal{X}$.
2. Normalisation: ${\sum }_{x\in \mathcal{X}}{p}_X(x)=1$.

For every $B\subseteq \mathcal{X}$, the probability that $X$ takes a value in $B$ is

$$P(X\in B)=\sum _{x\in B}{p}_X(x).$$

Why Event Probabilities Are Sums

The events $\{X=x\}$ for distinct values $x$ are pairwise disjoint. For any $B\subseteq \mathcal{X}$,

$$\{X\in B\}=\bigcup _{x\in B}\{X=x\}.$$

Countable additivity of $P$ therefore gives

$$P(X\in B)=\sum _{x\in B}P(X=x)=\sum _{x\in B}{p}_X(x).$$

Taking $B=\mathcal{X}$ yields the normalisation condition ${\sum }_{x\in \mathcal{X}}{p}_X(x)=1$.

Contrast with a Probability Measure

Warning

A PMF and a probability measure have different inputs:

$$p_X(x)=P_X(\{x\}),P_X(B)=\sum _{x\in B}{p}_X(x).$$

The PMF $p_X$ assigns mass to an individual value $x$. The distribution $P_X$ assigns probability to a set of values $B$.

Contrast with a Probability Density Function

A PMF value $p_X(x)$ is itself the probability $P(X=x)$ and cannot exceed $1$. For a probability density function, probabilities are obtained by integration, and the density at one point is not itself a probability.

Example

Fair six-sided die

Let $X$ be the result of a fair die roll. Its PMF is

$$p_X(x)=\{\begin{array}{ll}\frac{1}{6},& x\in \{1,2,3,4,5,6\},\\ 0,& \text{otherwise}.\end{array}$$

The probability of an even result is obtained by summing the relevant masses:

$$P(X\text{ is even})=p_X(2)+p_X(4)+p_X(6)=\frac{1}{2}.$$