Chapter 5: Q.5.12 (page 215)

Use the identity of Theoretical Exercise 5.5 to derive E[X2] when X is an exponential random variable with parameter λ.

Short Answer

Expert verified

Thus,

$\int_{0}^{\infty} 2 x P (X > x) d x = \int_{0}^{\infty} 2 x e^{- λ x} d x = 2 \int_{0}^{\infty} x e^{- λ x} d x$

$Use partial integration . Let u = x and d v = e^{- λ x} . Hence, we have that d u = d x and v = - \frac{e^{- λ x}}{λ} . Thus$

$2 \int_{0}^{\infty} x e^{- λ x} d x = - {2 x \frac{e^{- λ x}}{λ} |}_{0}^{\infty} + \frac{2}{λ} \int_{0}^{\infty} e^{- λ x} d x = {\frac{2}{λ} \cdot (- \frac{e^{- λ x}}{λ}) |}_{0}^{\infty} = \frac{2}{λ^{2}}$

Step by step solution

Given Information

Derive E[X2] when X is an exponential random variable with parameter λ.

Explanation

From exercise 5, we have that.

$E (X^{2}) = \int_{0}^{\infty} 2 x P (X > x) d x$

$Now, we are given that X ~ Expo (λ) . Hence, we have that$

$P (X > x) = 1 - P (X \leq x) = 1 - (1 - e^{- λ x}) = e^{- λ x}$

Thus,

$\int_{0}^{\infty} 2 x P (X > x) d x = \int_{0}^{\infty} 2 x e^{- λ x} d x = 2 \int_{0}^{\infty} x e^{- λ x} d x$

$Use partial integration . Let u = x and d v = e^{- λ x} . Hence, we have that d u = d x and v = - \frac{e^{- λ x}}{λ} . Thus$

Hence, we have proved that

$E (X^{2}) = \frac{2}{λ^{2}}$

Final Answer

Thus,

$\int_{0}^{\infty} 2 x P (X > x) d x = \int_{0}^{\infty} 2 x e^{- λ x} d x = 2 \int_{0}^{\infty} x e^{- λ x} d x$

$Use partial integration . Let u = x and d v = e^{- λ x} . Hence, we have that d u = d x and v = - \frac{e^{- λ x}}{λ} . Thus$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use the identity of Theoretical Exercise 5.5 to derive E[X2] when X is an exponential random variable with parameter λ.

Short Answer

Step by step solution

Given Information

Explanation

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Decision Maths

Logic and Functions

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.