Probability Mass Function of a Discrete Random Variable

When a random variable $X$ is discrete, the way its probabilities are distributed over its possible values $x_1, x_2, \ldots, x_n$ is described by its probability mass function, written $p(x_i) = P(X = x_i)$, or simply $p_i$.

Conditions for a valid p.m.f.

For $p(x_i)$ to be a genuine probability mass function, it must satisfy two conditions:

In words: no probability can be negative, and the probabilities of all the possible values together must add up exactly to $1$, because the random variable is certain to take some value from its list.

Constructing a probability distribution table

To build the probability distribution of a discrete random variable from an experiment, three steps are followed:

1. List every possible value the random variable can take.
2. Find the probability of each value from the underlying sample space or given information.
3. Arrange the values and their probabilities in a table, and verify that the probabilities sum to $1$.

Illustration. Suppose two fair coins are tossed and $X$ denotes the number of heads obtained. The sample space is $\{HH, HT, TH, TT\}$, each outcome equally likely with probability $\tfrac{1}{4}$. Here $X$ can be $0$ (from $TT$), $1$ (from $HT$ or $TH$), or $2$ (from $HH$), giving the distribution:

Here every $p_i \geq 0$ and $\tfrac{1}{4} + \tfrac{1}{2} + \tfrac{1}{4} = 1$, so this is a valid probability distribution. A bar chart of this distribution, with $x_i$ on the horizontal axis and $p_i$ on the vertical axis, gives an immediate visual sense of which values are more likely — here the middle value $X=1$ is the most probable outcome. …