Q. 5.19 聽If X聽is an exponential random... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 5: Q. 5.19 (page 216)

If $X$ is an exponential random variable with a mean $1 / 位$ , show that
$E [X^{k}] = \frac{k!}{位^{k}} k = 1, 2, 鈥�$
Hint: Make use of the gamma density function to evaluate the preceding.

Short Answer

Expert verified

Using the equation of expectation, determine $E (X^{k})$ and the Gamma function within the integral.

Step by step solution

Find the Exponential variable.

We are provided that $X$ is an exponential random variable with a mean $\frac{1}{位}$ In other words, we have got that $X ~$ Expo $(位)$ By the idea about the mean of a function of a random variable quantity, we have that

localid="1649619285130" $E (X^{k}) = {鈭�}_{鈩�} x^{k} f_{X} (x) d x = {鈭�}_{0}^{鈭�} x^{k} 位 e^{- 位 x} d x$

Calculate the integral.

Let's calculate the integral, we've that

${鈭�}_{0}^{鈭�} x^{k} 位 e^{- 位 x} d x = 位 {鈭�}_{0}^{鈭�} x^{k} e^{- 位 x} d x$

Making substitutions $s = 位 x$ we've that

localid="1649619305189" $位 {鈭�}_{0}^{鈭�} x^{k} e^{- 位 x} d x = 位 {鈭�}_{0}^{鈭�} {(\frac{s}{位})}^{k} e^{- s} \frac{d s}{位} = \frac{1}{位^{k}} {鈭�}_{0}^{鈭�} s^{k} e^{- s} d s = \frac{螕 (k + 1)}{位^{k}}$

Finally, we've obtained that

localid="1649619316547" $E (X^{k}) = \frac{螕 (k + 1)}{位^{k}} = \frac{k!}{位^{k}}$

Which has been claimed.

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91影视

If $X$ is an exponential random variable with a mean $1 / 位$ , show that
$E [X^{k}] = \frac{k!}{位^{k}} k = 1, 2, 鈥�$
Hint: Make use of the gamma density function to evaluate the preceding.

Short Answer

Step by step solution

Find the Exponential variable.

Calculate the integral.

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

If Xis an exponential random variable with a mean1/位, show thatEXk=k!位kk=1,2,鈥�Hint: Make use of the gamma density function to evaluate the preceding.

Short Answer

Step by step solution

Find the Exponential variable.

Calculate the integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Applied Mathematics

Statistics

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

If $X$ is an exponential random variable with a mean $1 / 位$ , show that
$E [X^{k}] = \frac{k!}{位^{k}} k = 1, 2, 鈥�$
Hint: Make use of the gamma density function to evaluate the preceding.