Let X, and X, be independent exponential distributions with the same parameter 1. What is the joint distribution of Y, = X, + X2 and Y, X1 ? What are the distributions of Y, and X1+X2 Y,?
Let X, and X, be independent exponential distributions with the same parameter 1. What is the joint distribution of Y, = X, + X2 and Y, X1 ? What are the distributions of Y, and X1+X2 Y,?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
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Chapter1: Starting With Matlab
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Transcribed Image Text:Let \( X_1 \) and \( X_2 \) be independent exponential distributions with the same parameter \( \lambda \). What is the joint distribution of \( Y_1 = X_1 + X_2 \) and \( Y_2 = \frac{X_1}{X_1 + X_2} \)? What are the distributions of \( Y_1 \) and \( Y_2 \)?
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Step 1
Given: X1 & X2 are independent random variables with pdfs
&
Then their joint pdf f(x1, x2) (say) is given by
We are given
First, we find the joint density of
Let , then T is a bijection from . Now we need to find in terms of .
Multiplying Y1 & Y2 we get
then
So,
The Jacobian of transformation is
Thus the joint pdf of Y1 and Y2, g(y1, y2) is
Therefore, the marginal pdf of Y1 is
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