Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 7 / a

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

i

Solution

The

X_{i}

are independent Bernoulli random variables with success parameter

p

. Their sum has the binomial distribution $S_{n} \sim Bin (n, p)$ , whose probability mass function is

P (S_{n} = k) = (k n) p^{k} (1 - p)^{n - k}, k = 0, \dots, n .

Choose the

k

successful events in

(k n)

ways; independence gives the same product probability for each choice. At

p = 0

or

p = 1

, interpret the formula by its degenerate distribution or by continuity.

ii

Solution

An indicator random variable equals

1

with probability

p

and

0

with probability

1 - p

, so

E (X_{i}) = 1 \cdot p + 0 \cdot (1 - p) = p

. Also

X_{i}^{2} = X_{i}

pointwise, whence

E (X_{i}) = E (X_{i}^{2}) = p .

iii

Solution

By linearity of expectation,

E (S_{n}) = \sum_{i = 1}^{n} p = n p

. Part (ii) gives

Var (X_{i}) = E (X_{i}^{2}) - [E (X_{i})]^{2} = p - p^{2}

. Independent random variables have zero pairwise covariance, so the variance of their sum is the sum of their variances. Therefore

E (S_{n}) = n p, Var (S_{n}) = n p (1 - p) .

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