Solution Found!
Solved: Suppose that the random variables Y1 and Y2 have
Chapter 5, Problem 32E(choose chapter or problem)
Suppose that the random variables \(Y_{1}\) and \(Y_{2}\) have joint probability density function, \(f\left(y_{1}, y_{2}\right)\), given by (see Exercise 5.14)
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
6 y_{1}^{2} y_{2}, & 0 \leq y_{1} \leq y_{2}, y_{1}+y_{2} \leq 2, \\
0, & \text { elsewhere. }
\end{array}\right.
\)
a Show that the marginal density of \(Y_{1}\) is a beta density with \(\alpha=3\) and \(\beta=2\).
b Derive the marginal density of \(Y_{2}\).
c Derive the conditional density of \(Y_{2}\) given \(Y_{1}=y_{1}\).
d Find \(P\left(Y_{2}<1.1 \mid Y_{1}=.60\right)\).
Equation Transcription:
Text Transcription:
Y_1
Y_2
f(y_1,y_2)
f(y_1,y_2)=
6y_1^2 y_2, 0</=y_1</=y_2,y_1+y_2</=2,
0, elsewhere.
Y_1
alpha=3
beta=2
Y_2
Y_2
Y_1=y_1
P(Y_2<1.1|Y_1=.60)
Questions & Answers
QUESTION:
Suppose that the random variables \(Y_{1}\) and \(Y_{2}\) have joint probability density function, \(f\left(y_{1}, y_{2}\right)\), given by (see Exercise 5.14)
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
6 y_{1}^{2} y_{2}, & 0 \leq y_{1} \leq y_{2}, y_{1}+y_{2} \leq 2, \\
0, & \text { elsewhere. }
\end{array}\right.
\)
a Show that the marginal density of \(Y_{1}\) is a beta density with \(\alpha=3\) and \(\beta=2\).
b Derive the marginal density of \(Y_{2}\).
c Derive the conditional density of \(Y_{2}\) given \(Y_{1}=y_{1}\).
d Find \(P\left(Y_{2}<1.1 \mid Y_{1}=.60\right)\).
Equation Transcription:
Text Transcription:
Y_1
Y_2
f(y_1,y_2)
f(y_1,y_2)=
6y_1^2 y_2, 0</=y_1</=y_2,y_1+y_2</=2,
0, elsewhere.
Y_1
alpha=3
beta=2
Y_2
Y_2
Y_1=y_1
P(Y_2<1.1|Y_1=.60)
ANSWER:
Solution :
Step 1 of 4:
Let and
have joint density function.
Then the joint density function and
is
Our goal is:
a). We need to find the marginal density of is a beta density with
and
.
b). We need to derive the marginal density of .
c). We need to find derive the conditional density of .
d). We need to find the probability is less than 1.
a).
The marginal density function of a continuous random variable is
or
Where 0.
Therefore,
0, otherwise
The beta distribution with and
.
Then,
B()=
We substituteand
B()=
B()=
B()=
)
Therefore ).