Let Y be a random variable with probability density

Chapter 6, Problem 1E

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QUESTION:

Let \(Y\) be a random variable with probability density function given by

f(y)=\left\{\begin{array}{ll}2(1-y), & 0 \leq  y \leq 1 \\0, & \text { elsewhere }\end{array}\right.

a Find the density function of \(U_{1}=2 Y-1\)
b Find the density function of \(U_{2}=1-2 Y\).
c. Find the density function of \(U_{3}=Y^{2}\).
d Find \(E\left(U_{1}\right), E\left(U_{2}\right)\), and \(E\left(U_{3}\right)\) by using the derived density functions for these random variables.
e Find \(E\left(U_{1}\right), E\left(U_{2}\right)\), and \(E\left(U_{3}\right)\) by the methods of Chapter 4 .

Equation Transcription:

 {

Text Transcription:

Y

f(y)= {2(1-y),   0  \leq  y \leq 1  0,  elsewhere  

U1=2Y-1

U2=1-2Y

U3=Y2

E(U1),E(U2)

E(U3)

E(U1),E(U2)

E(U3)

Questions & Answers

QUESTION:

Let \(Y\) be a random variable with probability density function given by

f(y)=\left\{\begin{array}{ll}2(1-y), & 0 \leq  y \leq 1 \\0, & \text { elsewhere }\end{array}\right.

a Find the density function of \(U_{1}=2 Y-1\)
b Find the density function of \(U_{2}=1-2 Y\).
c. Find the density function of \(U_{3}=Y^{2}\).
d Find \(E\left(U_{1}\right), E\left(U_{2}\right)\), and \(E\left(U_{3}\right)\) by using the derived density functions for these random variables.
e Find \(E\left(U_{1}\right), E\left(U_{2}\right)\), and \(E\left(U_{3}\right)\) by the methods of Chapter 4 .

Equation Transcription:

 {

Text Transcription:

Y

f(y)= {2(1-y),   0  \leq  y \leq 1  0,  elsewhere  

U1=2Y-1

U2=1-2Y

U3=Y2

E(U1),E(U2)

E(U3)

E(U1),E(U2)

E(U3)

ANSWER:

Solution :

Step 1 of 5:

Let Y be a random variable with probability density function is given by

Our goal is:

a). We need to find the density function of .

b). We need to find the density function of .

c). We need to find .

d). We need to find E(), E() and E().

e). We need to find E(), E() and E() by the methods of chapter 4.

a).

Now we have to find the density function of .

Then we have to find .

=

=

=

Then the distribution function of is

.

And the density function of is


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