A random variable attaches a number to each outcome of a random process, such as the total when two dice are rolled.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Discrete and continuous
A discrete variable takes separate values, such as counts. A continuous one can take any value in a range, such as a height.
The distribution
A probability distribution lists every value with its probability. Those probabilities must total exactly 1.
The mean
The mean of a random variable is its expected value: each value times its probability, all added.
Notation
The mean of a random variable is written with the Greek letter mu, because it describes a whole population rather than a sample.
A useful check
If your probabilities do not total 1, an outcome has been missed or double counted.
A number attached to a random outcome
A random variable assigns a number to each outcome of a random process, which makes means and standard deviations available. It is the bridge from probability to statistical summaries.
The probability distribution
A list of values with their probabilities, which must sum to exactly 1. Checking that sum is a quick and reliable test that no outcome has been missed or double-counted.
Mean and standard deviation of a distribution
The mean is the expected value; the standard deviation measures how much outcomes typically deviate from it. Both are computed from the distribution, not from any observed sample.
Discrete and continuous
Discrete variables take separate values and have probabilities attached to each. Continuous ones take any value in a range, and probability attaches to intervals — the chance of exactly one value is zero.
Step 2: Try It Yourself
Tap and try it out.
Total 4
Total 5
Total 6
Total 7
Total 7 has the most. It has 3 more than Total 4.
Step 3: Watch an Example
One step at a time.
Watch Kofi Build a Distribution
Kofi tosses two coins and counts the heads.
- Step 1
The random variable takes the values 0, 1 and 2.
Step 4: Your Turn
Practice makes it stick.
The Total
Problem 1 of 2
A distribution has probabilities 0.2, 0.3 and one unknown. What must the third be?
The Mean
Problem 2 of 2
Values 0, 1, 2 with probabilities 0.25, 0.5, 0.25. What is the mean?
Distributions
1 of 8
Probabilities 0.1, 0.4 and one more. What must it be?
2 of 8
Values 1 and 3 with probabilities 0.5 each. Mean?
3 of 8
Two dice. How many ways to total 7?
4 of 8
Two dice. How many ways to total 2?
5 of 8
Number of siblings. Discrete or continuous? 1 discrete, 2 continuous.
6 of 8
Exact height. Discrete or continuous? 1 or 2.
7 of 8
Which must be true of a probability distribution?
8 of 8
Values 2 and 6 with probabilities 0.5 each. Mean?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Probabilities 0.3, 0.3 and one more. What must it be?
Question 2 of 2
What is a random variable?
What You Learned
- A random variable assigns a number to each random outcome.
- Its distribution lists every value with its probability, totalling 1.
- Its mean is the expected value: each value times its probability.