Topic 6 of 6 · Form 5
Further Probability and Statistics
Probability density functions; E(X) = ∫xf(x) dx; Var(X) = ∫x²f(x) dx − μ²; cumulative distribution functions; median and percentiles; distribution of functions of a random variable.
Notes for this topic 1 file, in order
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Continuous random variables
Probability density functions; E(X) = ∫xf(x) dx; Var(X) = ∫x²f(x) dx − μ²; cumulative distribution functions; median and percentiles; distribution of functions of a random variable.
Worked example: f(x) = 3x²/8 for 0 ≤ x ≤ 2. Find E(X).
∫₀² 3x³/8 dx = [3x⁴/32]₀² = 1.5
Inference using normal and t-distributions
Confidence intervals and hypothesis tests for a mean using the t-distribution (small samples, unknown variance); paired and two-sample tests; pooled variance.
Chi-squared tests
Goodness of fit (including binomial, Poisson, normal models) and contingency tables; degrees of freedom; combining small expected frequencies (< 5).
Non-parametric tests
Sign test, Wilcoxon signed-rank and rank-sum tests.
Probability generating functions
G(t) = E(tˣ); finding mean and variance from G′(1) and G″(1); sums of independent variables.
Check yourself
- How many degrees of freedom for a 3 × 4 contingency table?
- When is the t-distribution used instead of the normal?
- For a PGF G(t), what is E(X)?
Show answers
- (3 − 1)(4 − 1) = 6
- When the population variance is unknown and the sample is small (from a normal population).
- G′(1)
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