Probability Mass Functions

Definition

For a discrete random variable $X$, the probability mass function (PMF) gives the probability of each possible value:

$p(x) = P(X = x)$

The function $p$ completely describes the distribution of $X$.

Properties

A valid PMF satisfies:

1. Non-negativity: $p(x) \geq 0$ for all $x$

2. Sums to 1: $\sum_{x} p(x) = 1$ (over all possible values)

3. Probability computation: For any event $E$:

$P(X \in E) = \sum_{x \in E} p(x)$

Example: Fair Die

For a fair six-sided die:

$x$	$p(x)$
1	$1/6$
2	$1/6$
3	$1/6$
4	$1/6$
5	$1/6$
6	$1/6$

$p(x) = \frac{1}{6} \text{ for } x \in \{1, 2, 3, 4, 5, 6\}$

Example: Biased Coin

A biased coin shows heads with probability $p$. Let $X$ = number of heads in one flip.

$x$	$P(X = x)$
0	$1 - p$
1	$p$

The PMF: $p(0) = 1 - p$, $p(1) = p$.

Example: Two Dice Sum

Roll two fair dice. Let $X$ = sum of the dice. Values range from 2 to 12.

$x$	Outcomes	$p(x)$
2	(1,1)	$1/36$
3	(1,2), (2,1)	$2/36$
4	(1,3), (2,2), (3,1)	$3/36$
5	(1,4), (2,3), (3,2), (4,1)	$4/36$
6	…	$5/36$
7	…	$6/36$
8	$5/36$
9	$4/36$
10	$3/36$
11	$2/36$
12	$1/36$

Notice the symmetry around 7.

Computing with PMFs

To find $P(X \leq a)$:

$P(X \leq a) = \sum_{x \leq a} p(x)$

To find $P(a \leq X \leq b)$:

$P(a \leq X \leq b) = \sum_{x=a}^{b} p(x)$

Example

For the dice sum: $P(X \geq 10) = P(X = 10) + P(X = 11) + P(X = 12) = \frac{3}{36} + \frac{2}{36} + \frac{1}{36} = \frac{6}{36} = \frac{1}{6}$.

Support

The support of $X$ is the set of values where $p(x) > 0$:

$\text{supp}(X) = \{x : p(x) > 0\}$

For a fair die: $\text{supp}(X) = \{1, 2, 3, 4, 5, 6\}$.

Visualising PMFs

PMFs are often shown as bar charts, with bar heights representing probabilities. The total height of all bars equals 1.

Joint PMFs

For two discrete random variables $X$ and $Y$, the joint PMF is:

$p(x, y) = P(X = x, Y = y)$

It satisfies:

$\sum_{x} \sum_{y} p(x, y) = 1$

The marginal PMF of $X$ is:

$p_X(x) = \sum_{y} p(x, y)$

Summary

Concept	Formula
PMF definition	$p(x) = P(X = x)$
Normalisation	$\sum_x p(x) = 1$
Event probability	$P(X \in E) = \sum_{x \in E} p(x)$
Support	$\{x : p(x) > 0\}$