A-level Further Maths Year 13 · Topic 7
Further statistics
A level further statistics works with continuous distributions and estimation.
We will use probability density and distribution functions, the t-distribution and probability generating functions.
Then we meet non-parametric tests and test correlation and regression.
- 5 subtopics
- 261 questions
- Free: 7.1 notes and 10 questions
In this topic
- 7.1Continuous random variables: pdf, cdf, expectation and varianceFree below · 50 questions
- 7.2The t-distribution: hypothesis tests and confidence intervals for a meanOpen in Full Marks · 52 questions
- 7.3Probability generating functionsOpen in Full Marks · 50 questions
- 7.4Non-parametric tests: sign test and Wilcoxon testsOpen in Full Marks · 56 questions
- 7.5Correlation and regression: testing the PMCC and Spearman's rankOpen in Full Marks · 53 questions
7.1 · Free notes
Continuous random variables: pdf, cdf, expectation and variance
Key points
- A probability density function (pdf) f(x) satisfies f(x) ≥ 0 and ∫f(x) dx = 1 over the whole range; P(a < X < b) = ∫ₐᵇ f(x) dx.
- The cumulative distribution function (cdf) F(x) = P(X ≤ x) = ∫ f(t) dt from the lower end to x; f(x) = F′(x).
- Median m: F(m) = 0.5. Quartiles: F(q₁) = 0.25, F(q₃) = 0.75. Mode: where f is greatest.
- E(X) = ∫x f(x) dx, Var(X) = ∫x² f(x) dx − [E(X)]²; E(g(X)) = ∫g(x) f(x) dx.
- The continuous uniform (rectangular) distribution on [a, b]: f(x) = 1/(b − a), E(X) = (a + b)/2, Var(X) = (b − a)²/12.
In more detail
- Worked example: f(x) = (3/8)x² on 0 ≤ x ≤ 2. E(X) = ∫(3/8)x³ dx = 1.5; E(X²) = ∫(3/8)x⁴ dx = 2.4; Var(X) = 2.4 − 2.25 = 0.15.
- Piecewise pdfs: integrate each piece over its own interval and add; the cdf must be continuous.
- Exam tip: sketch the pdf first; it shows the mode and whether the mean is above or below the median.
Practice
Continuous random variables: pdf, cdf, expectation and variance: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. What must the total area under a pdf be?
- 1
- 0
- 0.5
- It varies
Show answer
A: 1. Total probability.
2. f(x) = kx² on 0 ≤ x ≤ 3. What is k?
- 1/3
- 9
- 1/9
- 1/27
Show answer
C: 1/9. k × 27/3 = 1 gives k = 1/9.
3. How is the pdf found from the cdf?
- Integrate
- Differentiate: f(x) = F′(x)
- Square it
- Take the reciprocal
Show answer
B: Differentiate: f(x) = F′(x). The cdf is the integral of the pdf.
4. How is the median m defined for a continuous random variable?
- f(m) = 0.5
- m = E(X)
- f(m) is maximum
- F(m) = 0.5
Show answer
D: F(m) = 0.5. Half the probability lies below m.
5. X ~ U(2, 8) (rectangular). What is E(X)?
- 5
- 6
- 3
- 10
Show answer
A: 5. (2 + 8)/2.
6. X ~ U(2, 8). What is Var(X)?
- 6
- 36
- 3
- 1
Show answer
C: 3. 36/12.
7. For f(x) = 2x on 0 ≤ x ≤ 1, what is E(X)?
- 1/2
- 2/3
- 1
- 1/3
Show answer
B: 2/3. ∫2x² dx = 2/3.
8. For f(x) = 2x on 0 ≤ x ≤ 1, what is the cdf F(x)?
- 2x
- x
- 1 − x²
- x²
Show answer
D: x². ∫₀ˣ 2t dt.
9. For f(x) = 2x on 0 ≤ x ≤ 1, what is the median?
- 1/√2
- 1/2
- 2/3
- 1/4
Show answer
A: 1/√2. x² = 0.5.
10. What is P(X = 0.5) for a continuous random variable?
- f(0.5)
- 0.5
- 0
- F(0.5)
Show answer
C: 0. A single point has zero probability.
Keep going in Full Marks
Full Marks has 50 questions on Continuous random variables: pdf, cdf, expectation and variance (40 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.