Let X and Y denote two independent Poisson random variables with and .
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![**Exercise 1.6**
Let \( X \) and \( Y \) denote two independent Poisson random variables with parameters \( \lambda \) and \( \mu \).
(a) Show that the random variable \( X + Y \) has the Poisson distribution with parameter \( \lambda + \mu \).
(b) Compute the conditional distribution \( \mathbb{P}(X = k \mid X + Y = n) \) given that \( X + Y = n \), for all \( k, n \in \mathbb{N} \).
(c) Assume that respective parameters of the distributions of \( X \) and \( Y \) are random, independent, and chosen according to an exponential distribution with parameter \( \theta > 0 \).
Give the probability distributions of \( X \) and \( Y \), and compute the conditional distribution \( \mathbb{P}(X = k \mid X + Y = n) \) given that \( X + Y = n \), for all \( k, n \in \mathbb{N} \).
(d) Assume now that \( X \) and \( Y \) have the same random parameter represented by a single exponentially distributed random variable \( \Lambda \) with parameter \( \theta > 0 \), independent of \( X \) and \( Y \).
Compute the conditional distribution \( \mathbb{P}(X = k \mid X + Y = n) \) given that \( X + Y = n \), for all \( k, n \in \mathbb{N} \).
This exercise guides students through understanding the properties and computations related to Poisson random variables, including finding resultant distributions from additive properties, conditional distributions, and the impact of parameter variability modeled by an exponential distribution. These steps illustrate the intertwining of Poisson and exponential distributions, underscoring key concepts in probability and stochastic processes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F75edf730-8a75-43bd-9e48-efe942a7f6d4%2Ffe1f1440-d8a0-472e-9ffc-e77634628bf5%2Fct7fqaw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 1.6**
Let \( X \) and \( Y \) denote two independent Poisson random variables with parameters \( \lambda \) and \( \mu \).
(a) Show that the random variable \( X + Y \) has the Poisson distribution with parameter \( \lambda + \mu \).
(b) Compute the conditional distribution \( \mathbb{P}(X = k \mid X + Y = n) \) given that \( X + Y = n \), for all \( k, n \in \mathbb{N} \).
(c) Assume that respective parameters of the distributions of \( X \) and \( Y \) are random, independent, and chosen according to an exponential distribution with parameter \( \theta > 0 \).
Give the probability distributions of \( X \) and \( Y \), and compute the conditional distribution \( \mathbb{P}(X = k \mid X + Y = n) \) given that \( X + Y = n \), for all \( k, n \in \mathbb{N} \).
(d) Assume now that \( X \) and \( Y \) have the same random parameter represented by a single exponentially distributed random variable \( \Lambda \) with parameter \( \theta > 0 \), independent of \( X \) and \( Y \).
Compute the conditional distribution \( \mathbb{P}(X = k \mid X + Y = n) \) given that \( X + Y = n \), for all \( k, n \in \mathbb{N} \).
This exercise guides students through understanding the properties and computations related to Poisson random variables, including finding resultant distributions from additive properties, conditional distributions, and the impact of parameter variability modeled by an exponential distribution. These steps illustrate the intertwining of Poisson and exponential distributions, underscoring key concepts in probability and stochastic processes.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 1 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![MATLAB: An Introduction with Applications](https://www.bartleby.com/isbn_cover_images/9781119256830/9781119256830_smallCoverImage.gif)
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
![Probability and Statistics for Engineering and th…](https://www.bartleby.com/isbn_cover_images/9781305251809/9781305251809_smallCoverImage.gif)
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
![Statistics for The Behavioral Sciences (MindTap C…](https://www.bartleby.com/isbn_cover_images/9781305504912/9781305504912_smallCoverImage.gif)
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
![MATLAB: An Introduction with Applications](https://www.bartleby.com/isbn_cover_images/9781119256830/9781119256830_smallCoverImage.gif)
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
![Probability and Statistics for Engineering and th…](https://www.bartleby.com/isbn_cover_images/9781305251809/9781305251809_smallCoverImage.gif)
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
![Statistics for The Behavioral Sciences (MindTap C…](https://www.bartleby.com/isbn_cover_images/9781305504912/9781305504912_smallCoverImage.gif)
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
![Elementary Statistics: Picturing the World (7th E…](https://www.bartleby.com/isbn_cover_images/9780134683416/9780134683416_smallCoverImage.gif)
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
![The Basic Practice of Statistics](https://www.bartleby.com/isbn_cover_images/9781319042578/9781319042578_smallCoverImage.gif)
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
![Introduction to the Practice of Statistics](https://www.bartleby.com/isbn_cover_images/9781319013387/9781319013387_smallCoverImage.gif)
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman