Probability Mass Function

For a discrete random variable $X$ taking values $x_1, x_2, x_3, \ldots$, the probability mass function (pmf), written $p(x)$ or $P(X = x)$, gives the probability with which $X$ takes each of its possible values. The complete list of values together with their probabilities is called the probability distribution of $X$, and is usually displayed as a table:

A function $p(x)$ qualifies as a valid pmf of a discrete random variable if and only if it satisfies two conditions:

1. Non-negativity: $p(x_i) \ge 0$ for every value $x_i$ that $X$ can take (a probability can never be negative).
2. Total probability: $\displaystyle\sum_{i} p(x_i) = 1$ (the random variable must take some value from its list with certainty).

If either condition fails, the table given is not a valid probability distribution.

Example. Two fair coins are tossed and $X$ = number of heads obtained. The sample space is $\{HH, HT, TH, TT\}$, each outcome equally likely with probability $\tfrac14$. Here $X$ can be $0, 1, 2$, and:

Check: each probability is $\ge 0$, and $\tfrac14 + \tfrac12 + \tfrac14 = 1$ — so this is a valid pmf. …