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 / b / i

Table of contents
- Solution i

Solution

Write

M_{n} = max_{i} E_{i}

. Each exponential distribution has cumulative distribution function

1 - e^{- x}

for

x \geq 0

. Independence therefore gives the maximum of independent exponential random variables formula

P (M_{n} \leq x) = {0, (1 - e^{- x})^{n}, x < 0, x \geq 0.

For any fixed

y

, the threshold

y + ln n

is nonnegative for all sufficiently large

n

. Then

P (M_{n} - ln n \leq y) = (1 - \frac{e ^{- y}}{n})^{n} ⟶ exp (- e^{- y}) .

This is convergence in distribution to the standard Gumbel distribution.

 Ancestors (13)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version