Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 26K / e

Table of contents
- Solution e

Solution

The displayed density is that of a gamma distribution with shape

k + 1

and rate one. Put

K = \sum_{i = 1}^{n} k_{i} .

By the additivity of independent gamma distributions with a common rate,

X_{1} + \dots + X_{n} \sim Gamma (K + n, 1) .

Equivalently, the independence of random variables makes the Laplace transform of a nonnegative random variable of the sum equal

\prod_{i = 1}^{n} (1 + s)^{- (k_{i} + 1)} = (1 + s)^{- (K + n)} .

Therefore the required probability density function is

f_{X_{1} + \dots + X_{n}} (x) = 1_{{x > 0}} \frac{x ^{K + n - 1} e ^{- x}}{( K + n - 1 )!} .

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