 Solution

ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2015/probability-and-computing/1/f/solution

For

X

B (n, p)

, where

n

is a nonnegative integer and

0 \leq p \leq 1

M_{X} (t) = E [e^{tX}] = k = 0 \sum n e^{t k} (k n) p^{k} (1 - p)^{n - k} = k = 0 \sum n (k n) (p e^{t})^{k} (1 - p)^{n - k} = (1 - p + p e^{t})^{n}, t \in R .

t

, since the random variable has finite support. The formula also includes the degenerate cases

p = 0

p = 1

and

n = 0

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