4.1. Suppose X and Y are independent random variables, which are jointly continuous. 1. Show carefully that the distribution of 'X + Y given X = x' is equal to the distribution of the random variable x +Y. Hint: Consider the change of variables h(x, y) = (x,x+y). 2. Suppose X and Y each have an exponential distribution with parameter a, find the joint p.d.f of X and X + Y. 3. In the situation of (ii) above calculate E[X | X + Y].
4.1. Suppose X and Y are independent random variables, which are jointly continuous. 1. Show carefully that the distribution of 'X + Y given X = x' is equal to the distribution of the random variable x +Y. Hint: Consider the change of variables h(x, y) = (x,x+y). 2. Suppose X and Y each have an exponential distribution with parameter a, find the joint p.d.f of X and X + Y. 3. In the situation of (ii) above calculate E[X | X + Y].
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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Step 1: Write the given information.
VIEWStep 2: Determine the density function X and Y along with transformation.
VIEWStep 3: Determine the joint probability distribution function of X and Y.
VIEWStep 4: Determine the joint probability distribution function of X and X + Y.
VIEWStep 5: Determine the conditional expectation of E[X|X+Y].
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