Question

1. Let X and Y be independent random variables, each having the exponential distribution with parameter A. Find the joint density function of X and X + Y, and deduce that the conditional density function of X, given that X + Y = a, is uniform on the interval (0, a) for each a > 0. In other words, the knowledge that X + Y = a provides no useful clue about the position of X in the interval (0, a).

          1. Let X and Y be independent random variables, each having the exponential distribution with parameter
A. Find the joint density function of X and X + Y, and deduce that the conditional density function
of X, given that X + Y = a, is uniform on the interval (0, a) for each a > 0. In other words, the
knowledge that X + Y = a provides no useful clue about the position of X in the interval (0, a).

1. Let X and Y be independent random variables, each having the exponential distribution with parameter
A. Find the joint density function of X and X + Y, and deduce that the conditional density function
of X, given that X + Y = a, is uniform on the interval (0, a) for each a > 0. In other words, the
knowledge that X + Y = a provides no useful clue about the position of X in the interval (0, a).

Added by Scott A.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Adi S

10/01/2023

Step 1

Step 1: Find the joint density function of X and X + Y Since X and Y are independent random variables with exponential distribution with parameter λ, their probability density functions are given by: f_X(x) = λ * e^(-λx) for x > 0 f_Y(y) = λ * e^(-λy) for y > Show more…

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Let x and Y be independent random variables, each having the exponential distribution with parameter lambda . Find the joint density function of x and x+Y, and deduce that the conditional density function of x, given that x+Y=a, is uniform on the interval (0,a) for each a>0. In other words, the knowledge that x+Y=a provides no useful clue about the position of x in the interval (0,a). 1. Let X and Y be independent random variables,each having the exponential distribution with parameter X. Find the joint density function of X and X + Y, and deduce that the conditional density function of X, given that X + Y = a, is uniform on the interval0,a) for each a > 0. In other words, the knowledge that X + Y = a provides no useful clue about the position of X in the interval (0,a)