01 Key Concepts
Random Variable (X)
A rule that assigns a numeric value to each outcome in a sample space, such as X = number of heads in 3 coin flips.
Discrete Random Variables
Take on a countable set of values, often whole numbers (number of defective items, number of customers).
Continuous Random Variables
Can take any value within a range, often from measurements (height, weight, time).
Probability Distribution
A table, formula, or graph showing every possible value of X and its associated probability.
Expected Value (Mean)
E(X) = sum of [x * P(x)] over all possible values — the long-run average value of X.
Variance and Standard Deviation
Variance measures how spread out the values of X are; standard deviation is the square root of variance, in the same units as X.
02 Key Formulas
- E(X) = sum of [x * P(x)]
- Var(X) = E(X^2) - [E(X)]^2
03 Solved Examples
Example 1 A coin is flipped twice. Let X = number of heads. Find the probability distribution of X.
- Possible outcomes: HH, HT, TH, TT, each with probability 1/4.
- X=0 (TT): P=1/4. X=1 (HT or TH): P=2/4. X=2 (HH): P=1/4.
Answer: P(X=0)=1/4, P(X=1)=1/2, P(X=2)=1/4
Example 2 Using the distribution above, find E(X).
- E(X) = 0*(1/4) + 1*(1/2) + 2*(1/4) = 0 + 0.5 + 0.5.
Answer: E(X) = 1
Example 3 A random variable X takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3. Find E(X).
- E(X) = 1*(0.2) + 2*(0.5) + 3*(0.3) = 0.2 + 1.0 + 0.9.
Answer: E(X) = 2.1
04 Practice Questions
1Is 'number of cars passing an intersection in an hour' discrete or continuous?
Discrete
2Is 'exact weight of a bag of rice' discrete or continuous?
Continuous
3X takes values 0,1 with P(0)=0.3, P(1)=0.7. Find E(X).
0.7
4A die is rolled; let X be the outcome. Find E(X).
3.5
5What does E(X) represent in plain terms?
The long-run average value of X over many repetitions
📄 Random Variables — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.