Notes › EENG 3421: Probability and Statistics for Engineers Lecture 7

Probability Mass Functions

calendar_today February 23, 2026 article 547 words access_time 4 min replay Modified February 23, 2026

Table of Contents

Random Variables

We now move from events in a sample space to numerical descriptions of outcomes
A random variable assigns a real number to each outcome of an experiment
The random variable is written with a capital letter such as $X$
The set of possible values of $X$ is called the range of $X$
We denote the range by $S_X$
If multiple random variables are used then we similarly write $S_Y$ and so on

Definition 07.1 (Random Variable).

A random variable consists of

An experiment with probability measure $P[\cdot]$ defined on a sample space $S$
A function that assigns a real number to each outcome in $S$

Equivalently

$$ X \colon S \to \mathbb R $$

Notation

Sometimes it is important to emphasize the mapping from sample outcomes to numerical values
We write $X(s)$ for the value assigned to outcome $s \in S$
The event $\{X = x\}$ means all sample points whose mapped value is $x$

Proposition 07.2 (Event Induced by a Random Variable Value).

For any real number $x$

$$ \{X = x\} = \{s \in S \mid X(s) = x\} $$

Discrete Random Variables

A discrete random variable is one whose range is countable
This means its possible values can be listed as a finite or countably infinite sequence

Definition 07.3 (Discrete Random Variable).

$X$ is a discrete random variable if the range of $X$ is a countable set

$$ S_X = \{x_1, x_2, \dots\} $$

In practice this lets us assign probabilities directly to individual values of $X$
These point probabilities are the building blocks for probabilities of larger events involving $X$

Probability Mass Function

For a discrete random variable the main object is the probability mass function
The PMF gives the probability that $X$ takes a specific value
It is a function defined on values in the range of $X$

Definition 07.4 (Probability Mass Function).

The probability mass function of a discrete random variable $X$ is

$$ P_X(x) = P[X = x] $$

The PMF can be written as a formula
The PMF can also be displayed as a table of values
A bar plot is also common when visualizing the probability masses
These are different representations of the same function

Theorem 07.5 (Basic Properties of a PMF).

Let $X$ be a discrete random variable with PMF $P_X(x)$ and range $S_X$

Nonnegativity

$$ P_X(x) \ge 0 \quad \text{for all } x $$

Total mass

$$ \sum_{x \in S_X} P_X(x) = 1 $$

Probability of an event in the range For any event $B \subseteq S_X$

$$ P[B] = \sum_{x \in B} P_X(x) $$

Corollary 07.6 (Probability of a Value Set).

If $B = \{x_{i_1}, x_{i_2}, \dots\}$ is any subset of $S_X$ then its probability is obtained by summing the corresponding masses

$$ P[X \in B] = \sum_{x \in B} P_X(x) $$

Corollary 07.7 (Complement in the Range).

For any $B \subseteq S_X$

$$ P[X \in B^c] = 1 - P[X \in B] $$

where $B^c$ is taken relative to $S_X$

Note

A candidate function is a valid PMF only if it satisfies

$P_X(x) \ge 0$ for all $x$
The total sum over the support equals $1$

If an unknown constant appears in a PMF formula then it is determined from

$$ \sum_{x \in S_X} P_X(x) = 1 $$

References

Course lecture slides 11–12

Sources

Course lecture slides 11–12

Graph

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