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Theorem 5.4. Let X be a continuous r.v. with p.d.f. fx(x). Let y = g(x) be strictly monotonic (increasing or decreasing) function of x. Assume that g(x) is differentiable (and hence continuous) for all x. Then the p.d.f. h(.) of the r.v. Y is given by : dx hy(y) = fx(x)| (5-22) %3D ... where x is expressed in terms of y, and the range of Y is determined from the given range of the variable X, on using the transformation y = g(x).

Theorem 5.4. Let X be a continuous r.v. with p.d.f. fx(x). Let y = g(x) be strictly monotonic (increasing or decreasing) function of x. Assume that g(x) is differentiable (and hence continuous) for all x. Then the p.d.f. h(.) of the r.v. Y is given by : dx hy(y) = fx(x)| (5-22) %3D ... where x is expressed in terms of y, and the range of Y is determined from the given range of the variable X, on using the transformation y = g(x).

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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Prove the theorem

Theorem 5-4. Let X be a continuous r.v. with p.d.f. fx(x). Let y = g(x) be strictly monotonic (increasing or decreasing) function of x. Assume that g(x) is differentiable (and hence continuous) for all x. Then the p.d.f. h(.) of the r.v. Y is given by : dx hy(y) = fx(x) dy · (5-22) .. where x is expressed in terms of y, and the range of Y is determined from the given range of the variable X, on using the transformation y = g(x).
Transcribed Image Text:Theorem 5-4. Let X be a continuous r.v. with p.d.f. fx(x). Let y = g(x) be strictly monotonic (increasing or decreasing) function of x. Assume that g(x) is differentiable (and hence continuous) for all x. Then the p.d.f. h(.) of the r.v. Y is given by : dx hy(y) = fx(x) dy · (5-22) .. where x is expressed in terms of y, and the range of Y is determined from the given range of the variable X, on using the transformation y = g(x).
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