 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability/probability/9/a/iii/solution

0 < F (x) < 1

and is smooth. Its derivative is

F^{'} (x) = e^{- x} exp (- e^{- x}) > 0,

so it is increasing and continuous, hence right-continuous. As

x \to - \infty

e^{- x} \to \infty

and

F (x) \to 0

x \to \infty

e^{- x} \to 0

and

F (x) \to 1

. Thus it has all the required cumulative distribution function properties, and defines the standard Gumbel distribution.

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