Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 9 / c / ii

Table of contents
- Solution ii

Solution

Differentiate the cumulative distribution function for

y > 0

using the chain rule and the standard normal density:

f_{Y} (y) = 2 Φ^{'} (2 y) \frac{1}{2 y} = \frac{2}{2 π 2 y} e^{- y} = \frac{e ^{- y}}{π y} .

Thus a probability density function is

f_{Y} (y) = ⎩ ⎨ ⎧ \frac{e ^{- y}}{π y}, 0, y > 0, y \leq 0.

The value assigned at zero is immaterial. The probability density function is unbounded near zero, but

F_{Y} (0) = 0

and continuity of the cumulative distribution function there show that there is no point mass. This is a gamma distribution with shape

1/2

and rate

1

.

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