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PDF of Matrix Transformation

by alpines4
Tags: matrix, transformation
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alpines4
#1
Mar2-12, 11:10 AM
P: 4
1. The problem statement, all variables and given/known data
Define X to be an n-vector of jointly continuous random variables X1, ..., Xn with joint pdf f(x) mapping R^n to R. Let A be an invertible nxn matrix and set Y=AX+b. I want to derive the pdf of f(y) in terms of f(x), the original pdf.



2. Relevant equations




3. The attempt at a solution

Given a random variable and its PDF f(x), the transformation of Y=g(X) is (given that g is one to one and thus has an inverse) f(g^{-1}(y)) * g'(y). I don't know how to generalize this to a matrix, however. I assume it will be kind of similar... Any help is appreciated. I just need some tips to get started. Thank you!
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sunjin09
#2
Mar4-12, 10:26 PM
P: 312
Quote Quote by alpines4 View Post
1. The problem statement, all variables and given/known data
Define X to be an n-vector of jointly continuous random variables X1, ..., Xn with joint pdf f(x) mapping R^n to R. Let A be an invertible nxn matrix and set Y=AX+b. I want to derive the pdf of f(y) in terms of f(x), the original pdf.



2. Relevant equations




3. The attempt at a solution

Given a random variable and its PDF f(x), the transformation of Y=g(X) is (given that g is one to one and thus has an inverse) f(g^{-1}(y)) * g'(y). I don't know how to generalize this to a matrix, however. I assume it will be kind of similar... Any help is appreciated. I just need some tips to get started. Thank you!

Intuitively the pdf of Y=g(X) is given by
[tex]f_X(x)|dx|=f_Y(y)|dy|[/tex]
In the case of vector x and y, dx and dy is related through the determinant of the Jacobian matrix between x and y


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