Let Y1 and Y2 have a joint density function given by a

QUESTION:

Let \(Y_{1} \text { and } Y_{2}\) have a joint density function given by

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

3 y_{1} & 0 \leq y_{2} \leq y_{1} \leq 1 \\

0, & \text { elsewhere }

\end{array}\right.\)

a Find the marginal density functions of  \(Y_{1} \text { and } Y_{2}\).
b Find \(P\left(Y_{1} \leq 3 / 4 \mid Y_{2} \leq 1 / 2\right)\).
c Find the conditional density function of \(Y_{1}\)  given \(Y_{2}=y_{2}\)
d Find \(P\left(Y_{1} \leq 3 / 4 \mid Y_{2}=1 / 2\right)\).

Equation Transcription:

Text Transcription:

Y_1 and Y_2

f(y_1, y_2)={3y_1   0 \leq  y_2 \leq  y_1 \leq 1  0  elsewhere    

Y_1 and Y_2

P(Y_ 1 \leq 3/4|Y_2 \leq 1/2)

Y_1

Y_2 and Y_2

P(Y_1 \leq 3/4|Y_2 \leq 1/2)