Past exam of the mathematics course of the University of Cambridge / 2003 / ia / Paper 2 / 3F (Probability)

Table of contents
- a 3F
  - Solution a
- b 3F
  - Solution b

a

Solution

For a nonnegative integer-valued random variable

X

, its probability generating function is

G_{X} (z) = E [z^{X}] = \sum_{k = 0}^{\infty} P (X = k) z^{k},

initially defined at least for

∣ z ∣ \leq 1

. A general real-valued random variable need not have a probability generating function in this sense. For a binomial distribution, the binomial theorem gives

G_{X} (z) = \sum_{k = 0}^{n} (k n) p^{k} (1 - p)^{n - k} z^{k} = (1 - p + p z)^{n} .

Differentiating the generating function produces factorial moments:

E X = G_{X}^{'} (1) = n p, E [X (X - 1)] = G_{X}^{''} (1) = n (n - 1) p^{2} .

Since

X^{2} = X (X - 1) + X

, the mean and variance are

E X = n p, Var (X) = n p (1 - p) .

These formulas include the degenerate endpoint probabilities and

n = 0

.

b

Solution

For independent random variables, their sum has the product of their probability generating functions:

G_{X + Y} (z) = E [z^{X} z^{Y}] = G_{X} (z) G_{Y} (z) = (1 - p + p z)^{n + m} .

Comparing polynomial coefficients identifies a binomial distribution with parameters

n + m, p

:

P (X + Y = k) = (k n + m) p^{k} (1 - p)^{n + m - k}, k = 0, \dots, n + m,

and the probability is zero at all other values. The common success probability is essential to this binomial form; independent binomial counts with different probabilities do not in general add to a binomial count.

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