2. Fix > 0, let X be a Poisson random variable (RV), i.e., X~ Po(A), and let Y = X{X>0} denote the RV X conditioned on the event {X>0}. Clearly, the support of Y is N = {1,2,3,...}. Please do the following: (a) (b) Derive the probability mass function of the RV Y, i.e., give py(k) in terms of k, A. Prove the PMF is valid, i.e., show that it sums to one. Hint: recall the series representation 0*/k!. et = -0 (c) Derive an expression for the expected value of Y, i.e., E[Y], in terms of A. Hint: use the total expectation theorem for E[X], conditioning on the partition {X>0} and {X=0}, recognizing E[Y] = E[X|X>0].
2. Fix > 0, let X be a Poisson random variable (RV), i.e., X~ Po(A), and let Y = X{X>0} denote the RV X conditioned on the event {X>0}. Clearly, the support of Y is N = {1,2,3,...}. Please do the following: (a) (b) Derive the probability mass function of the RV Y, i.e., give py(k) in terms of k, A. Prove the PMF is valid, i.e., show that it sums to one. Hint: recall the series representation 0*/k!. et = -0 (c) Derive an expression for the expected value of Y, i.e., E[Y], in terms of A. Hint: use the total expectation theorem for E[X], conditioning on the partition {X>0} and {X=0}, recognizing E[Y] = E[X|X>0].
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![2.
Fix > 0, let X be a Poisson random variable (RV), i.e., X~ Po(A), and let Y = X{X > 0}
denote the RV X conditioned on the event {X>0}. Clearly, the support of Y is N = {1,2,3,...}. Please
do the following:
(b)
e -
Derive the probability mass function of the RV Y, i.e., give py (k) in terms of k, X.
Prove the PMF is valid, i.e., show that it sums to one. Hint: recall the series representation
Σxk/k!.
Derive an expression for the expected value of Y, i.e., E[Y], in terms of A. Hint: use the
total expectation theorem for E[X], conditioning on the partition {X>0} and {X = 0}, recognizing
E[Y] = E[X|X>0].
Derive an expression for the expected squared value of Y, i.e., E[Y2], in terms of A. Hint:
use the total expectation theorem for E[X2] conditioning on the partition {X > 0} and {X
recognizing E[Y2] = E[X²|X > 0].
0},
=
(e)
Derive an expression for the variance of Y, i.e., var(Y), in terms of A. Hint: use var(Y)
E[Y²] - E[Y]² and substitute in the expressions from the previous two questions.
Prove that var(Y) < var(X). Hint: use the fundamental inequality 1- x < e for x ≥ 0.
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Transcribed Image Text:2.
Fix > 0, let X be a Poisson random variable (RV), i.e., X~ Po(A), and let Y = X{X > 0}
denote the RV X conditioned on the event {X>0}. Clearly, the support of Y is N = {1,2,3,...}. Please
do the following:
(b)
e -
Derive the probability mass function of the RV Y, i.e., give py (k) in terms of k, X.
Prove the PMF is valid, i.e., show that it sums to one. Hint: recall the series representation
Σxk/k!.
Derive an expression for the expected value of Y, i.e., E[Y], in terms of A. Hint: use the
total expectation theorem for E[X], conditioning on the partition {X>0} and {X = 0}, recognizing
E[Y] = E[X|X>0].
Derive an expression for the expected squared value of Y, i.e., E[Y2], in terms of A. Hint:
use the total expectation theorem for E[X2] conditioning on the partition {X > 0} and {X
recognizing E[Y2] = E[X²|X > 0].
0},
=
(e)
Derive an expression for the variance of Y, i.e., var(Y), in terms of A. Hint: use var(Y)
E[Y²] - E[Y]² and substitute in the expressions from the previous two questions.
Prove that var(Y) < var(X). Hint: use the fundamental inequality 1- x < e for x ≥ 0.
=
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