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 / i

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- Solution 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.

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