Problem # 1. Functions of random variables: a) Let random variable
x
have probability density function
f_(x)(x)=\alpha e^(-\alpha x)u(x)
, where
u(*)
is the unit step function, and
\alpha >0
. Define derived random variable
Y=g(x)=log_(e)[1-e^(-\alpha x)]
Find the
PDFf_(Y)(y)
. b) Next, consider any continuous random variable
W
with known
PDF,f_(W)(w)
. Define derived random variable
Z=h(W)=log_(e)[F_(W)(W)]
where
F_(W)(*)
is the
CDF
of
W
. Find the
PDFf_(Z)(z)
.

Question

Problem # 1. Functions of random variables: a) Let random variable

x

have probability density function

f_(x)(x)=\alpha e^(-\alpha x)u(x)

, where

u(*)

is the unit step function, and

\alpha >0

. Define derived random variable

Y=g(x)=log_(e)[1-e^(-\alpha x)]

Find the

PDFf_(Y)(y)

. b) Next, consider any continuous random variable

W

with known

PDF,f_(W)(w)

. Define derived random variable

Z=h(W)=log_(e)[F_(W)(W)]

where

F_(W)(*)

is the

CDF

of

W

. Find the

PDFf_(Z)(z)

.

Accepted Answer

Expert Answer to -  Problem # 1. Functions of random variables: a) Let random variable x have probability density funct

Answer

Solution for -  Problem # 1. Functions of random variables: a) Let random variable x have probability density funct

Answer

This an additional answer to -  Problem # 1. Functions of random variables: a) Let random variable x have probability density funct

(Solved): Problem # 1. Functions of random variables: a) Let random variable x have probability density funct ...