Cambridge International AS and A Level Mathematics 6: Probability and Statistics 2

Probability and Statistics 2 is the content for Cambridge International AS and A Level Mathematics 9709 Paper 6. It assumes Paper 5 statistics and Paper 3 calculus. The five official sections cover Poisson models, linear combinations of random variables, continuous random variables, sampling and estimation, and hypothesis tests.

1. The Poisson distribution

A Poisson model Po(m) describes counts of random events in a fixed interval when events occur independently at a constant average rate and simultaneous occurrences are negligible. The parameter m is both the mean and variance. Use P$X = r$ = e^(-m)m^r/r! for r = 0, 1, 2 and so on.

Scale the mean with exposure. If a process averages 3 events per hour, then a two-hour count has mean 6 under the same model assumptions. State the interval attached to m.

A Poisson distribution can approximate B(n, p) when n is large and p is small. Cambridge gives the approximate guide n greater than 50 and np less than 5. Use m = np. A normal distribution can approximate Po(m) when m is large, approximately at least 15, with mean and variance m and a continuity correction.

2. Linear combinations of random variables

Expectation is linear: E(aX + b) = aE(X) + b, and E(aX + bY) = aE(X) + bE(Y). Variance is unaffected by adding a constant and scales by the square of a multiplier: Var(aX + b) = a^2 Var(X).

For independent X and Y, Var(aX + bY) = a^2 Var(X) + b^2 Var(Y). The plus between variance contributions remains even when the random variables are subtracted, because the coefficient is squared. Without independence, this formula is not justified by the Paper 6 result.

A linear transformation of a normal variable remains normal. A linear combination of independent normal variables is normal. The sum of independent Poisson variables is Poisson with parameter equal to the sum of their parameters. A difference of Poisson variables does not inherit that result.