Further Statistics: Discrete Random Variables and Expectation
A discrete random variable X takes a countable set of distinct numerical values, each occurring with a probability given by P(X=x), where every probability is non-negative and the probabilities across all possible values sum to 1. Further Statistics moves beyond the single named distributions of A Level Maths to general discrete random variables defined by a table or a formula, and introduces the algebra of expectation: E(X) = sum of x*P(X=x) is the long-run average value, E(X^2) is used to find the variance Var(X) = E(X^2) - [E(X)]^2, and linear coding gives E(aX+b) = aE(X) + b and Var(aX+b) = a^2*Var(X). The topic also covers the discrete uniform distribution on the integers 1 to n, where E(X) = (n+1)/2 and Var(X) = (n^2-1)/12.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Check the probability distribution is valid before doing anything else: list every value of X with its probability, confirm each P(X=x) >= 0, and confirm the probabilities sum to 1 - use this equation to find any unknown constant.
- Find E(X) = sum of x*P(X=x) over every possible value of X; this weighted average is the mean of the distribution.
- Find E(X^2) = sum of x^2*P(X=x), then use Var(X) = E(X^2) - [E(X)]^2 rather than trying to build a new table of (x - mean)^2 values.
- For a linear coding Y = aX + b, use E(Y) = aE(X) + b and Var(Y) = a^2*Var(X) directly - never rebuild the whole probability table for Y unless the question specifically asks for its distribution.
- For the discrete uniform distribution on the integers 1 to n (each value equally likely), quote E(X) = (n+1)/2 and Var(X) = (n^2-1)/12, or derive them from the standard sums of x and x^2 from 1 to n if the question asks for a derivation.
- Check the final answer makes sense: E(X) should lie between the smallest and largest value X can take, and Var(X) must always be positive.
Worked example
The random variable X has probability distribution given in the table below, where P(X=3) = k. x: 1, 2, 3, 4. P(X=x): 1/10, 3/10, k, 1/4. (a) Find k. (b) Find E(X) and Var(X). (c) Given that Y = 3X - 2, find E(Y).
- Use sum P(X=x) = 1: 1/10 + 3/10 + k + 1/4 = 1. Writing every fraction as twentieths gives 2/20 + 6/20 + k + 5/20 = 1.
- Solve for k: 13/20 + k = 1, so k = 7/20.
- Find E(X) = sum x*P(X=x) = 1(2/20) + 2(6/20) + 3(7/20) + 4(5/20) = (2+12+21+20)/20 = 55/20 = 11/4.
- Find E(X^2) = sum x^2*P(X=x) = 1(2/20) + 4(6/20) + 9(7/20) + 16(5/20) = (2+24+63+80)/20 = 169/20.
- Use Var(X) = E(X^2) - [E(X)]^2 = 169/20 - (11/4)^2 = 169/20 - 121/16 = 676/80 - 605/80 = 71/80.
- Use E(aX+b) = aE(X)+b for part (c): E(Y) = 3(11/4) - 2 = 33/4 - 8/4 = 25/4. Final answer: k = 7/20, E(X) = 11/4, Var(X) = 71/80, E(Y) = 25/4.
Practice questions
Try each question, then tap to reveal the answer.
Q1The discrete random variable Y has P(Y=1) = 0.2, P(Y=2) = 0.5, P(Y=3) = 0.3. Find E(Y).Show answer
Answer: E(Y) = 1(0.2) + 2(0.5) + 3(0.3) = 2.1
Q2For the random variable Y above, find Var(Y).Show answer
Answer: Var(Y) = E(Y^2) - [E(Y)]^2 = (0.2+2.0+2.7) - 2.1^2 = 4.9 - 4.41 = 0.49
Q3State the two conditions a set of values P(X=x) must satisfy to define a valid probability distribution for a discrete random variable X.Show answer
Answer: Every P(X=x) must be greater than or equal to 0, and the probabilities across all possible values of X must sum to 1.
Q4A discrete random variable W is uniformly distributed on the integers 1 to 9 inclusive. Find E(W) and Var(W).Show answer
Answer: E(W) = (9+1)/2 = 5; Var(W) = (9^2-1)/12 = 80/12 = 20/3
Q5Given that E(X) = 4 and Var(X) = 3, find E(2X+5) and Var(2X+5).Show answer
Answer: E(2X+5) = 2(4)+5 = 13; Var(2X+5) = 2^2(3) = 12
Q6A discrete random variable T has E(T) = 6 and E(T^2) = 40. Find Var(T).Show answer
Answer: Var(T) = 40 - 6^2 = 40 - 36 = 4
Q7A fair six-sided die is rolled once and X is the score obtained. Use the discrete uniform distribution to find E(X).Show answer
Answer: E(X) = (6+1)/2 = 7/2 = 3.5
Exam-style questions
Written in the style of a A Level Further Maths exam paper, with a full mark scheme.
A discrete random variable X has probability distribution P(X=x) = kx for x = 1, 2, 3, 4, and P(X=x) = 0 for any other value of x. (a) Show that k = 1/10. (b) Find E(X).
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The random variable X has probability distribution: x: -1, 0, 1, 2. P(X=x): 0.2, 0.3, p, 0.1. (a) Find p. (b) Find Var(X). (c) Given that Y = 4X - 3, find Var(Y).
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The random variable X has probability distribution defined by P(X=x) = (5-x)/10 for x = 1, 2, 3, 4, and P(X=x) = 0 otherwise. (a) Show that this is a valid probability distribution. (b) Find E(X) and Var(X).
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