Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Discrete mathematics, probability and continuous mathematics / 7 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

For a discrete random variable taking nonnegative integer values, its probability generating function is

G_{X} (s) = E [s^{X}] = k = 0 \sum \infty P (X = k) s^{k} .

It is defined at least for

∣ s ∣ \leq 1

. The usual probability generating function is intended for nonnegative integer-valued random variables; a general discrete real-valued random variable need not have this form. The coefficient of

s^{k}

recovers

P (X = k)

.

ii

Solution

For the binomial distribution, the binomial theorem gives

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

Differentiation of the probability generating function recovers the first expected value and second factorial moment:

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

Since

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

, the variance is

G_{X}^{''} (1) + G_{X}^{'} (1) - G_{X}^{'} (1)^{2} = n (n - 1) p^{2} + n p - n^{2} p^{2} = n p (1 - p) .

Thus

G_{X} (s) = (1 - p + p s)^{n}, E [X] = n p, var (X) = n p (1 - p) .

The formulas also hold for

n = 0, 1

and the degenerate choices

p = 0, 1

, with derivatives read directly from the resulting polynomial.

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