Probability mass function

The probability mass function of a probability measure $P$ on a countable set $S$ , equipped with all its subsets, is the function

p:S\to[0,1],\qquad p(x)=P(\{x\}).

Characterization

The masses satisfy $\sum_{x\in S}p(x)=1$ , and $P(A)=\sum_{x\in A}p(x)$ for every $A\subseteq S$ . Conversely, any nonnegative function on $S$ whose sum is one defines a probability measure by this formula.

For a random variable $X$ taking values in $S$ , its mass function is that of its law: $p_X(x)=\mathbb P(X=x)$ .

Example

A Bernoulli distribution with parameter $\theta$ has masses $p(1)=\theta$ and $p(0)=1-\theta$ .

Mass versus density

A mass is a point probability. A density with respect to Lebesgue measure gives probabilities by integration, not by evaluating the density at a point.