Past exam of the mathematics course of the University of Cambridge / 2002 / ia / Paper 2 / 11F / a

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- Solution a

Solution

For independent random variables, the probability density function of a sum is the convolution of probability densities. Since each density is one on

[0, 1]

and zero elsewhere,

f_{X + Y} (u) = \int_{R} 1_{[0, 1]} (x) 1_{[0, 1]} (u - x) d x .

This is the length of

[0, 1] \cap [u - 1, u]

. Therefore the density is

f_{X + Y} (u) = ⎩ ⎨ ⎧ u 2 - u 0 0 \leq u \leq 1, 1 \leq u \leq 2, otherwise .

The two triangular areas sum to one, as a probability density function must.

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