Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 7 / a / i

Table of contents
- Solution i

Solution

For the exponential distribution with rate

θ

, the change of variables

u = θ x

gives

E [X] = \frac{1}{θ} \int_{0}^{\infty} u e^{- u} d u = \frac{1}{θ}, E [X^{2}] = \frac{1}{θ ^{2}} \int_{0}^{\infty} u^{2} e^{- u} d u = \frac{2}{θ ^{2}} .

The integrals are one and two by integration by parts. Hence

E [X] = 1/ θ, Var (X) = 1/ θ^{2}

.

 Ancestors (12)

 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