Question

Let Y-ar+b (a) Find the mean and variance of Y in terms of the mean and variance of X b) Evaluate the mean and variance ofY if Xhas the following PDF: (a)-ele (c) Evaluate the mean and variance of Y if Xis the Gaussian random variable with mean 0 and variance d) Evaluate the mean and variance of Yif X-bcos 2U) where U is a uniform random variable in of 1 the unit interval.

Answer 1

(a)

Using properties of expectation and variance, and

(b)

Given fx(x) =-e-Ill i.e fx(x) Cedr +/4ur-if.re. +「re-j-o Now. EA]

$\begin{align*} E[X^2]&=\int_{-\infty}^0 x^2. \dfrac{1}{2}e^x dx +\int_0^{\infty} x^2. \dfrac{1}{2}e^{-x} dx=\dfrac{1}{2}\left[\int_{-\infty}^0 x^2e^{x}+\int_0^{\infty} x^2e^{-x}\right]\\ &=2 \end{align*}$

ELX Hence, from part (a)

(c)

From part (a) EM-aElX) + b = b