Unit 2 · Topic 2.8 · about 20 minutes

Introduction to Random Variables and Probability Distributions

Define a discrete random variable in context and construct its probability distribution, including its cumulative distribution, from the rules of probability or from a simulation.

Predict first

Roll two fair dice and let X be the larger of the two numbers (if they match, X is that number). Which value of X is the most likely?

Random variables

A random variable is a variable whose values are numbers produced by a random process. It is named with a capital letter, like $X$ , and its possible values are written in lowercase, like $x$ . The larger of two dice is a random variable. So are the number of free throws a player makes in a game and the number of texts you get in the next hour.

This lesson is about discrete random variables, which take a countable set of values such as 0, 1, 2 and so on. Variables that can take any value in an interval, like a time or a height, come in 2.11 The Normal Distribution.

Probability distributions

A probability distribution for a discrete random variable gives the probability of every possible value. Every probability in it is between 0 and 1, and the probabilities add to exactly 1.

A distribution can be worked out with the rules of probability, or estimated with a simulation. It can be shown as a table, as a graph, or as a function, meaning a formula that gives the probability for each value.

Worked exampleThe distribution of the larger die

Two fair dice are rolled, and X is the larger of the two numbers. Construct the probability distribution of X.

List the possible values. X can be 1, 2, 3, 4, 5 or 6.
Count the outcomes for each value. All 36 ordered pairs are equally likely. X = 1 only for (1, 1). X = 2 for (1, 2), (2, 1) and (2, 2), three pairs. Each step up adds two more pairs, so the counts are 1, 3, 5, 7, 9 and 11.
Divide each count by 36. The probabilities are $\frac{1}{36}, \frac{3}{36}, \frac{5}{36}, \frac{7}{36}, \frac{9}{36}$ and $\frac{11}{36}$ .
Check the total. $1 + 3 + 5 + 7 + 9 + 11 = 36$ , so the probabilities add to $\frac{36}{36} = 1$ , as every distribution must.

Answer.

The table below. As a function, $P (X = x) = \frac{2 x - 1}{36}$ for $x = 1, 2, \dots, 6$ .

Probability distribution of X, the larger of two dice
$x$	1	2	3	4	5	6
$P (X = x)$	$\frac{1}{36}$	$\frac{3}{36}$	$\frac{5}{36}$	$\frac{7}{36}$	$\frac{9}{36}$	$\frac{11}{36}$
As a decimal	0.028	0.083	0.139	0.194	0.250	0.306

The same distribution as a graph

Each bar's height is the probability of that value of X, and the six heights add to 1.

Cumulative probabilities

A cumulative probability distribution gives, for each value, the probability that X is less than or equal to that value. Build it by adding probabilities from the left: $P (X \leq 3) = \frac{1}{36} + \frac{3}{36} + \frac{5}{36} = \frac{9}{36} = 0.25$ .

For the larger die the cumulative probabilities are square numbers over 36, so as a function, $P (X \leq x) = \frac{x^{2}}{36}$ . The grid of rolls from 2.4 Introduction to Probability shows why: $X \leq 3$ means both dice show 3 or less, which is a 3-by-3 block of 9 cells.

Cumulative distribution of X, the larger of two dice
$x$	1	2	3	4	5	6
$P (X \leq x)$	$\frac{1}{36}$	$\frac{4}{36}$	$\frac{9}{36}$	$\frac{16}{36}$	$\frac{25}{36}$	$\frac{36}{36}$

A cumulative table answers "at most" questions directly and "at least" questions through the complement: $P (X \geq 5) = 1 - P (X \leq 4) = 1 - \frac{16}{36} = \frac{20}{36} \approx 0.556$ . The larger die is 5 or 6 more than half the time.

Estimating a distribution by simulation

Some distributions are easier to simulate than to work out. In basketball's one-and-one, a player shoots one free throw. A miss ends the trip with 0 points. A make earns a second shot, so the trip ends with 1 or 2 points. Take a 70% shooter whose shots are independent, and let X be the points she scores. A simulation of 1,000 trips to the line gave the counts below.

The rules of probability give the exact distribution to compare against: $P (X = 0) = 0.30$ , $P (X = 1) = 0.70 (0.30) = 0.21$ and $P (X = 2) = 0.70 (0.70) = 0.49$ . Each simulated relative frequency lands within 0.02 of the exact value.

One-and-one for a 70% shooter: 1,000 simulated trips against the exact distribution
Points scored, $x$	0	1	2	Total
Simulated count	292	228	480	1,000
Relative frequency	0.292	0.228	0.480	1
Exact probability	0.30	0.21	0.49	1

Lab

Binomial Explorer

The distributions in this lab belong to one family, the binomial, which is coming in 2.10 The Binomial Distribution. For now, use the lab to read a distribution as a graph: each bar is one value of X, the heights always add to 1, and choosing P(X ≤ k) under Find shades "at most k", a cumulative probability.

Open the full Binomial Explorer lab

Check your understanding

Each list gives probabilities for the values 0, 1, 2 and 3 of a random variable X, in that order. Which list is a valid probability distribution?

Let X be the number of cars owned by a randomly chosen household in a town. Its distribution is below, with one probability missing.

\begin{matrix} x & 0 & 1 & 2 & 3 & 4 \\ P (X = x) & 0.08 & 0.35 & ? & 0.17 & 0.06 \end{matrix}

Find $P (X \geq 2)$ . Give a decimal.

Your answer

The number of siblings X of a randomly chosen student at a school has this cumulative distribution:

\begin{matrix} x & 0 & 1 & 2 & 3 \\ P (X \leq x) & 0.20 & 0.55 & 0.85 & 1 \end{matrix}

What is $P (X = 2)$ ?

Let X be the number of goals a soccer team scores in a randomly chosen game, with $P (X = 0) = 0.25$ , $P (X = 1) = 0.35$ , $P (X = 2) = 0.25$ and $P (X = 3) = 0.15$ . What is $P (X < 2)$ ?

A player who makes 60% of her free throws goes to the line for a one-and-one. If she makes the first shot she takes a second; if she misses the first, the trip is over. Her shots are independent. Let X be the number of points she scores. What is $P (X = 1)$ ?

Course alignment, for teachers

AP Statistics topic 2.8, Unit 2: Probability, Random Variables, and Probability Distributions.

Skill 3.A: Construct tabular and graphical representations of data and distributions.