Solution Found!
In Exercise 5.9, we determined that is a valid joint
Chapter 5, Problem 27E(choose chapter or problem)
In Exercise 5.9, we determined that
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
6\left(1-y_{2}\right), & 0 \leq y_{1} \leq y_{2} \leq 1, \\
0, & \text { elsewhere }
\end{array}\right.
\)
is a valid joint probability density function. Find
a the marginal density functions for \(Y_{1}\) and \(Y_{2}\).
b \(P\left(Y_{2} \leq 1 / 2 \mid Y_{1} \leq 3 / 4\right)\).
c the conditional density function of \(Y_{1}\) given \(Y_{2}=y_{2}\).
d the conditional density function of \(Y_{2}\) given \(Y_{1}=y_{1}\).
e \(P\left(Y_{2} \geq 3 / 4 \mid Y_{1}=1 / 2\right)\).
Equation Transcription:
Text Transcription:
f(y_1,y_2)=
6(1-y_2), 0</=y_1</=y_2</=1,
0, elsewhere
Y_1
Y_2
P(Y_2</=1/2|Y_1</=3/4)
Y_1
Y_2=y_2
Y_2
Y_1=y_1
P(Y_2>/=3/4|Y_1=1/2)
Questions & Answers
QUESTION:
In Exercise 5.9, we determined that
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
6\left(1-y_{2}\right), & 0 \leq y_{1} \leq y_{2} \leq 1, \\
0, & \text { elsewhere }
\end{array}\right.
\)
is a valid joint probability density function. Find
a the marginal density functions for \(Y_{1}\) and \(Y_{2}\).
b \(P\left(Y_{2} \leq 1 / 2 \mid Y_{1} \leq 3 / 4\right)\).
c the conditional density function of \(Y_{1}\) given \(Y_{2}=y_{2}\).
d the conditional density function of \(Y_{2}\) given \(Y_{1}=y_{1}\).
e \(P\left(Y_{2} \geq 3 / 4 \mid Y_{1}=1 / 2\right)\).
Equation Transcription:
Text Transcription:
f(y_1,y_2)=
6(1-y_2), 0</=y_1</=y_2</=1,
0, elsewhere
Y_1
Y_2
P(Y_2</=1/2|Y_1</=3/4)
Y_1
Y_2=y_2
Y_2
Y_1=y_1
P(Y_2>/=3/4|Y_1=1/2)
ANSWER:
Answer:
Step 1 of 5:
(a)
We have given the joint probability density function.
We need to find the marginal density functions for and
.
Let and
jointly continuous random variables with the joint (or bivariate) probability function
Then the marginal density functions of
and
respectively, are given by
Hence the marginal density functions for is,
Since limit given is hence we can write the limit of
is
The marginal density functions for is,
Since limit given is hence we can write the limit of
is
Hence the marginal density functions of and
respectively, are given by