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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![Certainly! Here’s a transcription of the text suitable for an educational website:
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**Problem 2:**
Fix \( \lambda > 0 \), let \( X \) be a Poisson random variable (RV), i.e., \( X \sim \text{Po}(\lambda) \), and let \( Y \overset{\Delta}{=} X \mid \{X > 0\} \) denote the RV \( X \) conditioned on the event \(\{X > 0\}\). Clearly, the support of \( Y \) is \( \mathbb{N} = \{1, 2, 3, \ldots\}\). Please do the following:
(a) **Derive the probability mass function** of the RV \( Y \), i.e., give \( p_Y(k) \) in terms of \( k, \lambda \).
(b) **Prove the PMF is valid**, i.e., show that it sums to one. Hint: recall the series representation
\[
e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}.
\]
(c) **Derive an expression for the expected value of \( Y \)**, i.e., \(\mathbb{E}[Y]\), in terms of \( \lambda \). Hint: use the total expectation theorem for \(\mathbb{E}[X]\), conditioning on the partition \(\{X > 0\}\) and \(\{X = 0\}\), recognizing \(\mathbb{E}[Y] = \mathbb{E}[X \mid X > 0]\).
(d) **Derive an expression for the expected squared value of \( Y \)**, i.e., \(\mathbb{E}[Y^2]\), in terms of \( \lambda \). Hint: use the total expectation theorem for \(\mathbb{E}[X^2]\), conditioning on the partition \(\{X > 0\}\) and \(\{X = 0\}\), recognizing \(\mathbb{E}[Y^2] = \mathbb{E}[X^2 \mid X > 0]\).
(e) **Derive an expression for the variance of \( Y \)**, \(\text{var}(Y)\), in terms of \( \lambda \). Hint](/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:Certainly! Here’s a transcription of the text suitable for an educational website:
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**Problem 2:**
Fix \( \lambda > 0 \), let \( X \) be a Poisson random variable (RV), i.e., \( X \sim \text{Po}(\lambda) \), and let \( Y \overset{\Delta}{=} X \mid \{X > 0\} \) denote the RV \( X \) conditioned on the event \(\{X > 0\}\). Clearly, the support of \( Y \) is \( \mathbb{N} = \{1, 2, 3, \ldots\}\). Please do the following:
(a) **Derive the probability mass function** of the RV \( Y \), i.e., give \( p_Y(k) \) in terms of \( k, \lambda \).
(b) **Prove the PMF is valid**, i.e., show that it sums to one. Hint: recall the series representation
\[
e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}.
\]
(c) **Derive an expression for the expected value of \( Y \)**, i.e., \(\mathbb{E}[Y]\), in terms of \( \lambda \). Hint: use the total expectation theorem for \(\mathbb{E}[X]\), conditioning on the partition \(\{X > 0\}\) and \(\{X = 0\}\), recognizing \(\mathbb{E}[Y] = \mathbb{E}[X \mid X > 0]\).
(d) **Derive an expression for the expected squared value of \( Y \)**, i.e., \(\mathbb{E}[Y^2]\), in terms of \( \lambda \). Hint: use the total expectation theorem for \(\mathbb{E}[X^2]\), conditioning on the partition \(\{X > 0\}\) and \(\{X = 0\}\), recognizing \(\mathbb{E}[Y^2] = \mathbb{E}[X^2 \mid X > 0]\).
(e) **Derive an expression for the variance of \( Y \)**, \(\text{var}(Y)\), in terms of \( \lambda \). Hint
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