Solution Found!
Suppose that Y1 is the total time between a customer’s
Chapter 5, Problem 33E(choose chapter or problem)
Suppose that \(Y_{1}\) is the total time between a customer’s arrival in the store and departure from the service window, \(Y_{2}\) is the time spent in line before reaching the window, and the joint density of these variables (as was given in Exercise 5.15) is
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
e^{-y_{1}}, & 0 \leq y_{2} \leq y_{1} \leq \infty, \\
0, & \text { elsewhere. }
\end{array}\right.
\)
a Find the marginal density functions for \(Y_{1}\) and \(Y_{2}\).
b What is the conditional density function of \(Y_{1}\) given that \(Y_{2}=y_{2}\)? Be sure to specify the values of \(y_{2}\) for which this conditional density is defined.
c What is the conditional density function of \(Y_{2}\) given that \(Y_{1}=y_{1}\)? Be sure to specify the values of \(y_{1}\) for which this conditional density is defined.
d Is the conditional density function \(f\left(y_{1} \mid y_{2}\right)\) that you obtained in part (b) the same as the marginal density function \(f_{1}\left(y_{1}\right)\) found in part (a)?
e What does your answer to part (d) imply about marginal and conditional probabilities that
\(Y_{1}\) falls in any interval?
Equation Transcription:
Text Transcription:
Y_1
Y_2
f(y_1,y_2)=
e^-y_1, 0</=y_2</=y_1</=infinity,
0, elsewhere.
Y_1
Y_2
Y_1
Y_2=y_2
y_2
Y_2
Y_1=y_1
y_1
f(y_1|y_2)
f_1(y_1)
Y_1
Questions & Answers
QUESTION:
Suppose that \(Y_{1}\) is the total time between a customer’s arrival in the store and departure from the service window, \(Y_{2}\) is the time spent in line before reaching the window, and the joint density of these variables (as was given in Exercise 5.15) is
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
e^{-y_{1}}, & 0 \leq y_{2} \leq y_{1} \leq \infty, \\
0, & \text { elsewhere. }
\end{array}\right.
\)
a Find the marginal density functions for \(Y_{1}\) and \(Y_{2}\).
b What is the conditional density function of \(Y_{1}\) given that \(Y_{2}=y_{2}\)? Be sure to specify the values of \(y_{2}\) for which this conditional density is defined.
c What is the conditional density function of \(Y_{2}\) given that \(Y_{1}=y_{1}\)? Be sure to specify the values of \(y_{1}\) for which this conditional density is defined.
d Is the conditional density function \(f\left(y_{1} \mid y_{2}\right)\) that you obtained in part (b) the same as the marginal density function \(f_{1}\left(y_{1}\right)\) found in part (a)?
e What does your answer to part (d) imply about marginal and conditional probabilities that
\(Y_{1}\) falls in any interval?
Equation Transcription:
Text Transcription:
Y_1
Y_2
f(y_1,y_2)=
e^-y_1, 0</=y_2</=y_1</=infinity,
0, elsewhere.
Y_1
Y_2
Y_1
Y_2=y_2
y_2
Y_2
Y_1=y_1
y_1
f(y_1|y_2)
f_1(y_1)
Y_1
ANSWER:
Solution :
Step 1 of 5:
Let and
have joint density function.
Then the joint density function and
is
Our goal is:
a). We need to find the marginal density function of and
.
b). We need to find the conditional function of and
.
c). We need to is it the conditional function that we obtained in part (b) the same as the
same same as the marginal density function found in part (a).
d). We need to find what does your to part (d) imply about marginal and conditional probability
that falls in any interval.
a).
Now we have to find the marginal density function of and
.
The marginal density function of a continuous random variable is
or
For is
Therefore, .
And
is
Therefore, .