Let X be a Poisson(1) random variable, and let Y be the random variable distributed as Uniform(0, X). (a) Compute EY . (b) Compute Corr(X, Y ). (c) Determine a formula for the conditional probability density fX|Y (x|y) of X given Y . (d) What is the distribution of the conditional expectation E[X|Y ]? What about E[Y |X]?

Let X be a Poisson(1) random variable, and let Y be the random variable distributed as Uniform(0, X). (a) Compute EY . (b) Compute Corr(X, Y ). (c) Determine a formula for the conditional probability density fX|Y (x|y) of X given Y . (d) What is the distribution of the conditional expectation E[X|Y ]? What about E[Y |X]?

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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Question

Let X be a Poisson(1) random variable, and let Y be the random variable distributed as

Uniform(0, X).

(a) Compute EY .

(b) Compute Corr(X, Y ).

(d) What is the distribution of the conditional expectation E[X|Y ]? What about E[Y |X]?

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Step 4: Determine distribution of the conditional expectation

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