Problem 4: Suppose the random variable X in Problem 3 is sent through aquant izer g(r) to produce the random variable Y = g(X), wherex < -0.5,0, -0.5 <* < 0.5,x 2 0.5.-1,g(x)(a) Find a mathematical expression for Fy (y), the CDF of Y.(b) Find a mathematical expression for fy (y), the PDF of Y.(c) Find the value of E[Y].

a) For the given F(x) function, we have here:Fx(-0.5) = 0.2*(1 - 0.5)2 = 0.05Fx(0.5) = 0.5*(1 + 0.5)…

Answered: Problem 4: Suppose the random variable…

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

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Problem 4: Suppose the random variable X in Problem 3 is sent through a
quant izer g(r) to produce the random variable Y = g(X), where
x < -0.5,
0, -0.5 <* < 0.5,
x 2 0.5.
-1,
g(x)
(a) Find a mathematical expression for Fy (y), the CDF of Y.
(b) Find a mathematical expression for fy (y), the PDF of Y.
(c) Find the value of E[Y].

Expert Solution

Step 1

a) For the given F(x) function, we have here:
F_x(-0.5) = 0.2*(1 - 0.5)² = 0.05
F_x(0.5) = 0.5*(1 + 0.5) = 0.75

Therefore, the CDF for Y here is given as:

$F_y(y) = \left\{\begin{matrix} 0 & y < -1 \\ 0.05 & -1 \leq y < 0\\ 0.75 - 0.05 = 0.7 & 0 \leq y < 1 \\ 1 - 0.75 = 0.25 & y\geq 1 \end{matrix}\right.$

This is the required CDF here.

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