Probability Mass Function (PMF)

Learn what a Probability Mass Function (PMF) is, how it represents probabilities for discrete random variables, its properties, and simple examples with tables.

1. Meaning of a Probability Mass Function

A Probability Mass Function (PMF) gives the probability that a discrete random variable takes a particular value. It tells us how the total probability is distributed across the possible numerical values of the variable.

If X is a discrete random variable, then a PMF assigns a number to each possible value of X:

\( P(X = x) \)

This value shows how likely it is for X to be equal to x.

2. Properties of a PMF

A PMF must follow certain rules:

  • Non-negative: For every possible value x,

    \( P(X = x) \ge 0 \)

  • Total probability is 1:

    \( \sum P(X = x) = 1 \)

  • Defined only for discrete values: PMFs apply only to variables that take separate, countable values.

3. Example 1: PMF of a Coin Toss

Let X = 1 if Head appears, and X = 0 if Tail appears.

XP(X=x)
01/2
11/2

Each value has probability 1/2, and the total is 1.

4. Example 2: PMF of Rolling a Die

Let X be the number appearing on the die. Each number from 1 to 6 has equal chance.

XP(X=x)
11/6
21/6
31/6
41/6
51/6
61/6

5. Example 3: A Discrete Custom PMF

Let X be the number of heads in two coin tosses. The possible values are {0, 1, 2}.

The PMF is:

XP(X=x)
01/4
11/2
21/4

This PMF shows how probability is distributed among the three possible values.