Question

Let $Y_1$ and $Y_2$ have the joint density function $\f(y_1, y_2) = \begin{cases} e^{-(y_1 + y_2)}, & y_1 > 0, y_2 > 0, \\ 0, & \text{elsewhere.} \end{cases}$ (a) Are $Y_1$ and $Y_2$ independent? $\circ$ $Y_1$ and $Y_2$ are independent because $f(y_1, y_2) = f_1(y_1)f_2(y_2)$. $\circ$ $Y_1$ and $Y_2$ are dependent because $f(y_1, y_2) = f_1(y_1)f_2(y_2)$. $\circ$ $Y_1$ and $Y_2$ are independent because $f(y_1, y_2) \ne f_1(y_1)f_2(y_2)$. $\circ$ $Y_1$ and $Y_2$ are dependent because $f(y_1, y_2) \ne f_1(y_1)f_2(y_2)$. (b) Using the information from part (a), can we conclude that the marginal density function $f_1(y_1)$ will be equal to the conditional density function $f(y_1|y_2)$? $Y_1$ and $Y_2$ are ---Select--- if and only if $f_1(y_1) = f(y_1|y_2)$. The result in (a) indicates that $f_1(y_1)$ ---Select--- be equal to $f(y_1|y_2)$.

          Let $Y_1$ and $Y_2$ have the joint density function
$\f(y_1, y_2) = \begin{cases} e^{-(y_1 + y_2)}, & y_1 > 0, y_2 > 0, \\ 0, & \text{elsewhere.} \end{cases}$

(a) Are $Y_1$ and $Y_2$ independent?
$\circ$ $Y_1$ and $Y_2$ are independent because $f(y_1, y_2) = f_1(y_1)f_2(y_2)$.
$\circ$ $Y_1$ and $Y_2$ are dependent because $f(y_1, y_2) = f_1(y_1)f_2(y_2)$.
$\circ$ $Y_1$ and $Y_2$ are independent because $f(y_1, y_2) \ne f_1(y_1)f_2(y_2)$.
$\circ$ $Y_1$ and $Y_2$ are dependent because $f(y_1, y_2) \ne f_1(y_1)f_2(y_2)$.

(b) Using the information from part (a), can we conclude that the marginal density function $f_1(y_1)$ will be equal to the conditional density function $f(y_1|y_2)$?
$Y_1$ and $Y_2$ are ---Select--- if and only if $f_1(y_1) = f(y_1|y_2)$. The result in (a) indicates that $f_1(y_1)$ ---Select--- be equal to $f(y_1|y_2)$.

Let Y1 and Y2 have the joint density function
(y1, y2) = e^-(y1 + y2), y1 > 0, y2 > 0,
0, elsewhere.

(a) Are Y1 and Y2 independent?
∘ Y1 and Y2 are independent because f(y1, y2) = f1(y1)f2(y2).
∘ Y1 and Y2 are dependent because f(y1, y2) = f1(y1)f2(y2).
∘ Y1 and Y2 are independent because f(y1, y2) f1(y1)f2(y2).
∘ Y1 and Y2 are dependent because f(y1, y2) f1(y1)f2(y2).

(b) Using the information from part (a), can we conclude that the marginal density function f1(y1) will be equal to the conditional density function f(y1|y2)?
Y1 and Y2 are —Select— if and only if f1(y1) = f(y1|y2). The result in (a) indicates that f1(y1) —Select— be equal to f(y1|y2).

Added by Ryan M.

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