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].
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
Related questions
Question
![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.
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7ffb148-54c7-4dce-915c-269933801524%2F63128315-8b74-4243-80dd-4d76f18cb790%2Fldigbdm_processed.jpeg&w=3840&q=75)
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.
=
Expert Solution
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 4 steps
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,