Let Y1 and Y2 have joint density function first

QUESTION:

Let \(Y_{1}\) and \(Y_{2}\) have joint density function first encountered in Exercise 5.7:

                          \(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}

e^{-\left(y_{1}+y_{2}\right)}, & y_{1}>0, y_{2}>0, \\

0, & \text { elsewhere. }

\end{array}\right.

\)

a Find the marginal density functions for \(Y_{1}\) and \(Y_{2}\). Identify these densities as one of those studied in Chapter 4.

b What is \(P\left(1<Y_1<2.5\right)?\ P\left(1<Y_2<2.5\right)\)?

c For what values of \(y_{2}\) is the conditional density \(f\left(y_{1} \mid y_{2}\right)\) defined?

d For any \(y_{2}>0\), what is the conditional density function of \(Y_{1}\) given that \(Y_{2}=y_{2}\)?

e For any \(y_{1}>0\), what is the conditional density function of \(Y_{2}\) given that \(Y_{1}=y_{1}\)?

f For any\(y_{2}>0\), how does the conditional density function \(f\left(y_{1} \mid y_{2}\right)\) that you obtained in part (d) compare to the marginal density function \(f_{1}\left(y_{1}\right)\) found in part (a)?

g What does your answer to part (f) 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+y_2), y1>0,y2>0,

0, elsewhere.

Y_1

Y_2

P(1<Y1<2.5)? P(<Y2<2.5)

y_2

f(y_1|y_2)

y_2>0

Y_1

Y_2=y_2

y_1>0

Y_2

Y_1=y_1

y_2>0

f(y_1|y_2)

f_1(y_1)

Y_1