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Let X and Y denote two independent Poisson random variables with and .

Let X and Y denote two independent Poisson random variables with and .

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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**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.
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.
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