Probability and Statistics for Engineers and Scientists
Probability and Statistics for Engineers and Scientists
9th Edition
ISBN: 9780321629111
Author: Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying Ye
Publisher: Prentice Hall
bartleby

Concept explainers

bartleby

Videos

Question
Book Icon
Chapter 17.6, Problem 1RE
To determine

Prove that the random variable i=1nXi is a Poisson random variable with mean i=1nμi and variance i=1nμi.

Expert Solution & Answer
Check Mark

Explanation of Solution

Calculation:

According to Theorem 7.10, it is known that for n random variables X1,X2,....,Xn, the moment generating function can be written as follows:

MX1+X2+....+Xn=MX1(t)MX2(t)...MXn(t), where MXi(t)'s are moment generating function of random variable Xi for all i=1,2,...,n.

The moment generating function of Poisson random variable X with mean and variance μ is known as MX(t)=eμ(et1).

According to given question X1,X2,...,Xn are independent Poison random variables with means μ1,μ2,....,μn, respectively.

Now, using Theorem 7.10, the moment generating function of i=1nXi is obtained as follows:

Mi=1nXi=MX1+X2+....+Xn=MX1(t)MX2(t)...MXn(t)=eμ1(et1)eμ2(et1)....eμn(et1)=e(μ1+μ2+...+μ2)(et1)=e(i=1nμi)(et1)

Hence, it is proved that the random variable i=1nXi is a Poisson random variable with mean i=1nμi and variance i=1nμi.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Knowledge Booster
Background pattern image
Statistics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Text book image
MATLAB: An Introduction with Applications
Statistics
ISBN:9781119256830
Author:Amos Gilat
Publisher:John Wiley & Sons Inc
Text book image
Probability and Statistics for Engineering and th...
Statistics
ISBN:9781305251809
Author:Jay L. Devore
Publisher:Cengage Learning
Text book image
Statistics for The Behavioral Sciences (MindTap C...
Statistics
ISBN:9781305504912
Author:Frederick J Gravetter, Larry B. Wallnau
Publisher:Cengage Learning
Text book image
Elementary Statistics: Picturing the World (7th E...
Statistics
ISBN:9780134683416
Author:Ron Larson, Betsy Farber
Publisher:PEARSON
Text book image
The Basic Practice of Statistics
Statistics
ISBN:9781319042578
Author:David S. Moore, William I. Notz, Michael A. Fligner
Publisher:W. H. Freeman
Text book image
Introduction to the Practice of Statistics
Statistics
ISBN:9781319013387
Author:David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:W. H. Freeman
Continuous Probability Distributions - Basic Introduction; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=QxqxdQ_g2uw;License: Standard YouTube License, CC-BY
Probability Density Function (p.d.f.) Finding k (Part 1) | ExamSolutions; Author: ExamSolutions;https://www.youtube.com/watch?v=RsuS2ehsTDM;License: Standard YouTube License, CC-BY
Find the value of k so that the Function is a Probability Density Function; Author: The Math Sorcerer;https://www.youtube.com/watch?v=QqoCZWrVnbA;License: Standard Youtube License