A-level Further Maths Year 12 · Topic 7
Further statistics
Further statistics adds new distributions and tests.
We will work with discrete random variables, the Poisson, geometric and negative binomial distributions.
Then we use chi-squared tests and think about Type I and Type II errors.
- 5 subtopics
- 258 questions
- Free: 7.1 notes and 10 questions
In this topic
- 7.1Discrete random variables: expectation, variance and functions of XFree below · 49 questions
- 7.2The Poisson distribution in Further Statistics: sums and approximationsOpen in Full Marks · 51 questions
- 7.3Chi-squared tests: goodness of fit and contingency tablesOpen in Full Marks · 53 questions
- 7.4Type I and Type II errors in hypothesis testingOpen in Full Marks · 53 questions
- 7.5The geometric and negative binomial distributionsOpen in Full Marks · 52 questions
7.1 · Free notes
Discrete random variables: expectation, variance and functions of X
Key points
- A discrete random variable X takes values xᵢ with probabilities pᵢ, where Σpᵢ = 1.
- Expectation E(X) = Σxp. Variance Var(X) = E(X²) − [E(X)]², where E(X²) = Σx²p. Standard deviation = √Var(X).
- For constants a and b: E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).
- E(g(X)) = Σg(x)p for any function g; for example E(1/X) = Σ(1/x)p.
- The discrete uniform distribution on 1, 2, …, n has E(X) = (n + 1)/2 and Var(X) = (n² − 1)/12.
In more detail
- Worked example: X takes 0, 1, 2 with probabilities 0.25, 0.5, 0.25. E(X) = 1, E(X²) = 0.5 + 1 = 1.5, so Var(X) = 0.5.
- E(g(X)) ≠ g(E(X)) in general: for example E(X²) is not [E(X)]² unless the variance is zero.
- Exam tip: set out a table with columns x, P(X = x), xP and x²P to keep the calculations organised.
Practice
Discrete random variables: expectation, variance and functions of X: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. P(X = 1) = 0.2, P(X = 2) = 0.5, P(X = 3) = k. What is k?
- 0.5
- 0.2
- 0.7
- 0.3
Show answer
D: 0.3. Probabilities sum to 1.
2. For that distribution, what is E(X)?
- 2
- 2.1
- 1.5
- 2.5
Show answer
B: 2.1. 0.2 + 1 + 0.9.
3. E(X) = 4 and Var(X) = 2. What is E(3X − 1)?
- 11
- 12
- 13
- 5
Show answer
A: 11. 3 × 4 − 1.
4. E(X) = 4 and Var(X) = 2. What is Var(3X − 1)?
- 6
- 5
- 18
- 17
Show answer
C: 18. 9 × 2.
5. Which formula gives Var(X)?
- E(X²)
- [E(X)]² − E(X²)
- E(X) − E(X²)
- E(X²) − [E(X)]²
Show answer
D: E(X²) − [E(X)]². Mean of the squares minus square of the mean.
6. What is E(X) for a fair six-sided die?
- 3
- 3.5
- 6
- 21
Show answer
B: 3.5. (6 + 1)/2.
7. What is Var(X) for a fair six-sided die?
- 35/12
- 3.5
- 91/6
- 2
Show answer
A: 35/12. (36 − 1)/12.
8. E(X) = 2 and E(X²) = 7. What is Var(X)?
- 5
- 9
- 3
- 7
Show answer
C: 3. 7 − 4.
9. Does adding a constant change the variance?
- Yes, it adds the constant
- Yes, it doubles it
- Only if negative
- No
Show answer
D: No. Var(X + b) = Var(X).
10. What is E(X²) if X takes 1 and 3 with probability 0.5 each?
- 4
- 5
- 2
- 10
Show answer
B: 5. 0.5 + 4.5.
Keep going in Full Marks
Full Marks has 49 questions on Discrete random variables: expectation, variance and functions of X (39 more than here), set out in practice levels. You also get notes and 3D models for all 5 subtopics of Further statistics, plus games, timed exams and past papers.