To invite to a dinner, the White House randomly selects 9 athletes from a group of 5 runners, 7 gymnasts, and 6 boxers. Let X be the number of gymnasts invited, and Y the number of boxers invited. Find the joint probability mass function of X and Y and the conditional probability density function of X given Y=2.

Question

Quizplus · Accepted Answer

The Answer is (a)

p ( x , y ) = \left\{ \begin{array} { l l } \frac { \left( \begin{array} { l } 7 \x\end{array} ight) \left( \begin{array} { l } 6 \y\end{array} ight) \left( \begin{array} { c } 5 \0 - ( x + y )\end{array} ight) } { \left( \begin{array} { c } 18 \9\end{array} ight) } & x = 0,1 , \ldots , 7 , y = 0,1 , ext { dots } , 6,4 \leq x + y \leq 9 \0 & ext { otherwise. }\end{array} ight.

(b)

p _ { X \mid Y } ( x \mid 2 ) = \left\{ \begin{array} { l l } \frac { \left( \begin{array} { l } 7 \x\end{array} ight) \left( \begin{array} { l } 6 \2\end{array} ight) \left( \begin{array} { c } 5 \9 - ( x + 2 )\end{array} ight) } { \left( \begin{array} { c } 18 \9\end{array} ight) } & x = 0,1 \ldots , 7 \0 & ext { otherwise. }\end{array} ight.

To Invite to a Dinner, the White House Randomly Selects