Probability density function Let \( X \) have the pdf \( f(x) = \frac{e^{-x}}{(1+e^{-x})^2} \) with support \( S_X = (-\infty, \infty) \), and let \( Y = \frac{1}{1+e^{-X}} \). Show that for any \( a, b \in (0, 1) \), we have \( P(a < Y < b) = b - a \).

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Answered: Let X have the pdf f(x) e-x (1+e−x)²…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let X have the pdf f(x) e-x (1+e−x)² that for any a, b € (0, 1), we have P(a < Y < b) = b-a. = with support Sx = (-∞, ∞), and let y = 1 1+e-X• Show