Random Variable
In short: Random variable is a quantity whose numerical value is determined by the outcome of a chance process, described completely by listing its possible values and their probabilities.
AP StatisticsProbabilityAP Statistics Unit 2
What is Random Variable?
A random variable, usually named \(X\), converts every possible result of a chance process into a number. Heads in three tosses, dollars won at a carnival booth, and pets in a randomly picked household all qualify.
When the values can be listed, each one carries a probability, and the probabilities total 1. The mean is \(\mu_X = \sum x\,P(x)\), the long-run average over many repetitions, and the spread is \(\sigma_X = \sqrt{\sum (x – \mu_X)^2 P(x)}\).
How to spot it
- A chance process is described and a numerical result, such as a count or a payout, is attached to each outcome.
- A table lists values of X in one row and probabilities in the next.
- The question asks for the expected value or the standard deviation of a game.
- Phrases like “let X be the number of …” start the problem.
The method
- Define \(X\) in words and list every value it can take.
- Write the probability of each value and confirm that the probabilities sum to 1.
- Compute the mean by multiplying each value by its probability and adding.
- Compute the variance \(\sum (x – \mu_X)^2 P(x)\), then take the square root for \(\sigma_X\).
Random Variable: solved example
Problem
In a town survey, \(X\) counts the pets in a randomly chosen household, with \(P(0) = 0.30\), \(P(1) = 0.40\), \(P(2) = 0.20\), \(P(3) = 0.10\). Find the mean and standard deviation of \(X\).
Solution
Mean: \(0(0.30) + 1(0.40) + 2(0.20) + 3(0.10) = 0.40 + 0.40 + 0.30 = 1.1\).
Variance: \((0 – 1.1)^2(0.30) + (1 – 1.1)^2(0.40) + (2 – 1.1)^2(0.20) + (3 – 1.1)^2(0.10) = 0.363 + 0.004 + 0.162 + 0.361 = 0.89\).
Standard deviation: \(\sqrt{0.89} \approx\) 0.94. The average household has 1.1 pets, give or take about 0.94.
Try it
Question 1
For the pet distribution above, find \(P(X \ge 2)\).
Show the answer
\(0.20 + 0.10 = 0.30\).
Question 2
The outcomes of a spinner are worth 1, 2, 3 or 4 points, with chances 0.2, 0.5, \(k\) and 0.1. Find \(k\).
Show the answer
The probabilities must sum to 1: \(0.2 + 0.5 + k + 0.1 = 1\), so \(k = 0.2\).
Random Variable vs. Probability
A probability is one number between 0 and 1 attached to an event. A random variable is the numerical quantity itself, with a whole distribution of probabilities, a mean and a spread. The probability that \(X\) equals 2 is one entry in that distribution.
Keep going
Lessons
Worksheets and quizzes
- Random Variables, Expected Value and Combining Variables – Worksheet
- Random Variables, Expected Value and Combining Variables – Quiz
Related words
Whole dictionary: A–Z · AP Statistics · Probability