Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 8 / b

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii

i

Solution

For a nonnegative integer-valued random variable

X

, its probability generating function is

G_{X} (s) = E (s^{X}) = j = 0 \sum \infty P (X = j) s^{j}, 0 \leq s \leq 1.

At

s = 0

, use

0^{0} = 1

so

G_{X} (0) = P (X = 0)

. The series also converges absolutely for complex

∣ s ∣ \leq 1

; for

s

near zero, its coefficients recover the probability mass function.

ii

Solution

Summing a geometric series gives the probability generating function of the zero-based geometric distribution:

G_{N} (s) = j = 0 \sum \infty p (1 - p)^{j} s^{j} = \frac{p}{1 - ( 1 - p ) s} .

The series converges for

∣ (1 - p) s ∣ < 1

, in particular on

[0, 1]

. At

p = 1/2

,

G_{N} (s) = 1/ (2 - s)

,

G_{N}^{'} (s) = (2 - s)^{- 2}

, and

G_{N}^{''} (s) = 2 (2 - s)^{- 3}

. The factorial moment identities give

E (N) = G_{N}^{'} (1) = 1

and

E [N (N - 1)] = G_{N}^{''} (1) = 2

. Thus

E (N) = 1, Var (N) = G_{N}^{''} (1) + G_{N}^{'} (1) - [G_{N}^{'} (1)]^{2} = 2.

 Ancestors (11)

 View article source

 Discussion (0) New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version