 Solution

ID: past-exam-of-the-simulation-and-statistical-programming-course-of-the-university-of-oxford/2021/3/a/ii/solution

Solving

u = F (x)

gives

1 - (1 - p) e^{- β x} = p^{1 - u}

F^{- 1} (u) = - \frac{1}{β} lo g (\frac{1 - p ^{1 - u}}{1 - p}), 0 < u < 1.

For such

u

p < p^{1 - u} < 1

, making the logarithm's argument lie in

(0, 1)

and the output positive. Its endpoint limits are

F^{- 1} (0) = 0

and

F^{- 1} (1) = + \infty

u

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