About this chapter
Discrete random variables brings together probability and averages. A question gives a table or a formula for the probabilities, and asks for missing values, the mean, the variance, or the distribution of a new variable built from the old one. The last part is often a game, where you must list every way to win.
This chapter covers probability distributions and probability functions, cumulative distribution functions, E(X), E(X²) and Var(X), linear functions of a random variable, the discrete uniform distribution, finding unknown probabilities from two conditions, and combining independent observations.
The ten questions
- Q1 (5 marks): a probability function k(x+1): show k = 1/14, then an exact E(X) and a probability.
- Q2 (5 marks): two unknown probabilities from the total of 1 and a given E(X).
- Q3 (5 marks): recover a distribution from its cumulative distribution function.
- Q4 (5 marks): E(5 − 2X), Var(5 − 2X) and E(X²) from a given mean and variance.
- Q5 (5 marks): a fair 12-sided die: the uniform distribution, then Y = 4X − 3.
- Q6 (12 marks): a table mixing p(x) and F(x) with unknowns, show Var(X) = 1.46, then E(Y²) and two observations.
- Q7 (12 marks): a red and a blue four-sided die: sums, comparisons, and the distribution of the difference.
- Q8 (11 marks): a symmetric distribution, a uniform one, and a linear function W = aX + b fitted to both.
- Q9 (10 marks): points scored in a game: E(X) and Var(X), then the chance of reaching a prize total in the remaining rounds.
- Q10 (12 marks): a cumulative distribution function with k in it: show k = 2, the distribution, E(X), an inequality and Var(5 − 3X).
Key skills tested
Probability distributions. Every probability lies between 0 and 1 and they add to 1, which finds an unknown.
Probability functions. For a formula such as k(x+1), substitute each value of x, add, and set the total equal to 1.
Cumulative distribution. F(x) = P(X ≤ x), so P(X = x) = F(x) − F(x − 1). For whole-number X, F(2.7) equals F(2).
Expectation. E(X) = Σ x P(X = x), and E(X²) = Σ x² P(X = x).
Variance. Var(X) = E(X²) − [E(X)]². E(X²) is not the same as [E(X)]².
Linear functions. E(aX + b) = aE(X) + b, and Var(aX + b) = a² Var(X): the b disappears and the a is squared.
Discrete uniform. For n equally likely values 1 to n, E(X) = (n+1)/2 and Var(X) = (n² − 1)/12.
Unknown probabilities. The total of 1 and a given E(X) give two simultaneous equations.
Combining observations. Independent observations multiply. List every pair that gives the event; a sample space helps.
Worked example
Question 1 from this chapter. P(X = x) = k(x+1) for x = 1, 2, 3, 4. (a) Show that k = 1/14. (b) Find the exact value of E(X). (c) Find P(X ≥ 3).
(a) The probabilities are 2k, 3k, 4k and 5k. They add to 1, so 14k = 1 and k = 1/14.
(b) E(X) = (1 × 2 + 2 × 3 + 3 × 4 + 4 × 5) ÷ 14 = 40 ÷ 14 = 20/7.
(c) P(X ≥ 3) = P(X = 3) + P(X = 4) = (4 + 5) ÷ 14 = 9/14.
The question asks for an exact E(X), so 20/7 should stay a fraction rather than become 2.86.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2024, Q2: a distribution where E(X²) had to be shown before the variance, then a linear function aX + b and the outcomes of a game, as in Q6, Q8 and Q9.
- June 2022, Q5: two unknown probabilities from the total and E(X), a standard deviation, and games needing several probabilities multiplied, as in Q2 and Q9.
Where marks are lost
- a instead of a². The most common error in June 2024 was multiplying Var(X) by a rather than a² when finding the variance of aX + b.
- Skipping E(X²) in a show that. In June 2024 students lost marks for not showing E(X²) when asked to show a variance, since it could be worked backwards from the answer.
- Variance or standard deviation? In June 2022 many calculated the variance when the question asked for the standard deviation.
- The missing equation. In June 2022 many wrote the E(X) equation but not the sum of probabilities equal to 1, so could not find both unknowns.
- Too few probabilities multiplied. In June 2022 an event needed three probabilities multiplied, and most multiplied only two.
Common questions
Is E(X²) the same as [E(X)]²?
No. E(X²) squares each value before weighting; [E(X)]² squares the mean. Their difference is the variance.
Why does the constant disappear in Var(aX + b)?
Adding a constant shifts every value by the same amount, which moves the mean but does not change the spread.
How do I get probabilities from a cumulative distribution function?
Subtract consecutive values: P(X = x) = F(x) − F(x − 1).
Where this chapter leads
- Statistics 1 Chapter 7: The Normal Distribution: from discrete probabilities to a continuous model.
- Statistics 1 Chapter 4: Probability: the conditional probability used in the harder game questions.