Suppose that, for ?1 ? ? ? 1, the probability density

QUESTION:

Suppose that, for \(-1 \leq \alpha \leq 1\), the probability density function of \(\left(Y_1,\ Y_2\right)\) is given by

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

{\left[1-\alpha\left\{\left(1-2 e^{-y_{1}}\right)\left(1-2 e^{-y_{2}}\right)\right\}\right] e^{-y_{1}-y_{2}},} & 0 \leq y_{1}, 0 \leq y_{2}, \\

0, & \text { elsewhere. }

\end{array}\right.

\)

a Show that the marginal distribution of \(Y_{1}\) is exponential with mean 1.

b What is the marginal distribution of \(Y_{2}\)?

c Show that \(Y_{1}\) and \(Y_{2}\) are independent if and only if \(\alpha=0\).

Notice that these results imply that there are infinitely many joint densities such that both marginals are exponential with mean 1.

Equation Transcription:

Text Transcription:

-1</=alpha</=1

(Y_1,Y_2)

f(y_1,y_2)={_0,     elsewhere ^[1-alpha{(1-2e^-y_1)(1-2e^-y_2)}]e^-y_1-y_2, 0</=y1, 0</=y_2.

Y_1

Y_2

Y_1

Y_2

alpha=0