Unit 2 · Topic 2.8

2.8 Introduction to Random Variables and Probability Distributions

A random variable attaches a number to each outcome of a random process. For a discrete random variable, a probability distribution lists every value with its probability, and a cumulative distribution adds them up as you go.

Key terms

random variable
discrete random variable
probability distribution
cumulative probability

Random variables

A random variable is a numerical result of a random process. It's usually written with a capital letter like X. If you flip a coin 3 times and X = the number of heads, X can be 0, 1, 2 or 3.

A discrete random variable has values you can list: counts like 0, 1, 2, … . A continuous random variable can take any value in an interval, like a waiting time; you'll meet those in 2.11. For now, everything is discrete.

Probability distributions

A probability distribution gives the probability of each possible value. It can be a table, a graph (a histogram of probabilities) or a formula. Every probability must be between 0 and 1, and they must add to exactly 1.

For three fair coin flips, the 8 equally likely outcomes give:

Number of heads, x	0	1	2	3
P(X = x)	1/8	3/8	3/8	1/8
P(X ≤ x)	1/8	4/8	7/8	8/8

Cumulative probability

The cumulative distribution gives P(X ≤ x) for each value: add the probabilities from the smallest value up to x. The last entry is always 1.

Use it for "at most" questions. To get "more than," subtract from 1: P(X > 1) = 1 − P(X ≤ 1) = 1 − 4/8 = 1/2.

Graphing a distribution

A probability histogram puts the values of X on the horizontal axis and draws a bar above each value with height equal to its probability. You describe its shape, center and spread just as you would for data in Unit 1. The three-coin-flip distribution is symmetric, centered at 1.5 heads.

The difference from Unit 1: a data histogram shows what happened in one sample, while a probability histogram shows the model for what happens in the long run.

Where distributions come from

You can build a distribution with the probability rules (list outcomes, use the multiplication rule) or estimate one by simulation: run many trials and use the relative frequency of each value.

Pay attention to inequality words. "At least 2" means X ≥ 2. "More than 2" means X > 2. "At most 2" means X ≤ 2. "Fewer than 2" means X < 2. For a discrete variable, X < 2 and X ≤ 2 are different.

Worked examples

Try each one yourself first, then open the solution.

Example 1Calculator allowed
Find a missing probability and use it
Let X = the number of cars a randomly chosen household owns. P(X = 0) = 0.10, P(X = 1) = 0.35, P(X = 2) = 0.30, P(X = 3) = 0.15, and P(X = 4) is missing (no household owns more than 4). Find P(X = 4), P(X ≥ 2) and P(X < 2).
Show the solutionHide the solution
1. Step 1: Probabilities must add to 1: 0.10 + 0.35 + 0.30 + 0.15 = 0.90, so P(X = 4) = 0.10.
2. Step 2: P(X ≥ 2) = 0.30 + 0.15 + 0.10 = 0.55.
3. Step 3: P(X < 2) = P(X = 0) + P(X = 1) = 0.45. That's the complement of X ≥ 2, so 1 − 0.55 checks out.
Answer: P(X = 4) = 0.10, P(X ≥ 2) = 0.55, P(X < 2) = 0.45.
Example 2Calculator allowed
Trap: "at least" vs. "more than"
For three fair coin flips, a student finds P(more than 1 head) as 1 − P(X = 0) = 7/8. Find the correct answer.
Show the solutionHide the solution
1. Step 1: "More than 1" means X = 2 or X = 3.
2. Step 2: P(X > 1) = 3/8 + 1/8 = 4/8 = 1/2.
3. Step 3: The student calculated P(X ≥ 1), which includes X = 1.
Answer: 1/2. The student found "at least one," not "more than one."

Common mistakes

Mixing up ≥ and > (or ≤ and <) with discrete variables.
Building a distribution whose probabilities don't add to 1.
Listing outcomes (HTH, HHT, …) instead of values of the random variable (0, 1, 2, 3) in the distribution table.

On the exam

Expect to complete a probability distribution table, then use it to answer "at least" and "at most" questions.
If a question includes a histogram of a probability distribution, the bar heights are probabilities and should add to 1.

Connected topics

Videos

AP Stats 2.B.1 - Discrete Random Variables
Skew The ScriptWatch on YouTube (opens in a new tab)
AP Statistics Unit 2 | Random Variables & Probability Distributions | CED 2.8 | Guided Notes
Goldie's Math EmporiumWatch on YouTube (opens in a new tab)
AP Statistics: Topic 4.7 Introduction to Random Variables and Probability Distributions
Michael Porinchak - AP Statistics & AP PrecalculusWatch on YouTube (opens in a new tab)
Probability with discrete random variable example | Random variables | AP Statistics | Khan Academy
Khan AcademyWatch on YouTube (opens in a new tab)
An Introduction to Discrete Random Variables and Discrete Probability Distributions
jbstatisticsWatch on YouTube (opens in a new tab)
Valid discrete probability distribution examples | Random variables | AP Statistics | Khan Academy
Khan AcademyWatch on YouTube (opens in a new tab)