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*4. Let Y1, Y2 be random variables and h(Y1, Y2) be a transformation that is monotone in yı for each y2 on the joint support. Let h-l denote a marginal inverse transformation such that Y1 = h-1(u; y2). Show that the density of U = h(Y1,Y2) is: g(u) = | f (h- (u; y2), 92) | d -h'(u; y2) dyz du

*4. Let Y1, Y2 be random variables and h(Y1, Y2) be a transformation that is monotone in yı for each y2 on the joint support. Let h-l denote a marginal inverse transformation such that Y1 = h-1(u; y2). Show that the density of U = h(Y1,Y2) is: g(u) = | f (h- (u; y2), 92) | d -h'(u; y2) dyz du

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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**Educational Content:** *4. Let \( Y_1, Y_2 \) be random variables and \( h(y_1, y_2) \) be a transformation that is monotone in \( y_1 \) for each \( y_2 \) on the joint support. Let \( h^{-1} \) denote a marginal inverse transformation such that \( y_1 = h^{-1}(u; y_2) \). Show that the density of \( U = h(Y_1, Y_2) \) is: \[ g(u) = \int f\left(h^{-1}(u; y_2), y_2\right) \left| \frac{d}{du} h^{-1}(u; y_2) \right| dy_2 \]
Transcribed Image Text:**Educational Content:** *4. Let \( Y_1, Y_2 \) be random variables and \( h(y_1, y_2) \) be a transformation that is monotone in \( y_1 \) for each \( y_2 \) on the joint support. Let \( h^{-1} \) denote a marginal inverse transformation such that \( y_1 = h^{-1}(u; y_2) \). Show that the density of \( U = h(Y_1, Y_2) \) is: \[ g(u) = \int f\left(h^{-1}(u; y_2), y_2\right) \left| \frac{d}{du} h^{-1}(u; y_2) \right| dy_2 \]
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