Solution Found!
Let Y be a random variable with probability density
Chapter 6, Problem 1E(choose chapter or problem)
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