8.3 Problems On Random Vectors and Joint Distributions

Two cards are selected at random, without replacement, from a standard deck. Let X be the number of aces and Y be the number of spades. Under the usual assumptions, determine the joint distribution and the marginals.

Two positions for campus jobs are open. Two sophomores, three juniors, and three seniors apply. It is decided to select two at random (each possible pair equally likely). Let X be the number of sophomores and Y be the number of juniors who are selected. Determine the joint distribution for the pair $\left\{X , Y\right\}$ and from this determine the marginals for each.

As a variation of the exercise, Suppose a pair of dice is rolled instead of a single die. Determine the joint distribution for the pair $\left\{X , Y\right\}$ and from this determine the marginal distribution for Y.

Suppose a pair of dice is rolled. Let X be the total number of spots which turn up. Roll the pair an additional X times. Let Y be the number of sevens that are thrown on the X rolls. Determine the joint distribution for the pair $\left\{X , Y\right\}$ and from this determine the marginal distribution for Y. What is the probability of three or more sevens?

$f_{X Y} \left(t , u\right) = 1$ for $0 \leq t \leq 1$, $0 \leq u \leq 2 \left(1 - t\right)$.

$f_{X Y} \left(t , u\right) = 1 / 2$ on the square with vertices at $\left(1 , 0\right) , \left(2 , 1\right) , \left(1 , 2\right) , \left(0 , 1\right)$.

$f_{X Y} \left(t , u\right) = 4 t \left(1 - u\right)$ for $0 \leq t \leq 1$, $0 \leq u \leq 1$.

$f_{X Y} \left(t , u\right) = \frac{1}{8} \left(t + u\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq 2$.

$f_{X Y} \left(t , u\right) = 4 u e^{- 2 t}$ for $0 \leq t , 0 \leq u \leq 1$

$f_{X Y} \left(t , u\right) = \frac{3}{88} \left(2 t + 3 u^{2}\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq 1 + t$.

$f_{X Y} \left(t , u\right) = 12 t^{2} u$ on the parallelogram with vertices $\left(- 1 , 0\right) , \left(0 , 0\right) , \left(1 , 1\right) , \left(0 , 1\right)$

$f_{X Y} \left(t , u\right) = \frac{24}{11} t u$ for $0 \leq t \leq 2$, $0 \leq u \leq \min \left\{1 , 2 - t\right\}$

$f_{X Y} \left(t , u\right) = \frac{3}{23} \left(t + 2 u\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq \max \left\{2 - t , t\right\}$

$f_{X Y} \left(t , u\right) = \frac{12}{179} \left(3 t^{2} + u\right)$, for $0 \leq t \leq 2$, $0 \leq u \leq \min \left\{2 , 3 - t\right\}$

$f_{X Y} \left(t , u\right) = \frac{12}{227} \left(3 t + 2 t u\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq \min \left\{1 + t , 2\right\}$

$f_{X Y} \left(t , u\right) = \frac{2}{13} \left(t + 2 u\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq \min \left\{2 t , 3 - t\right\}$

$f_{X Y} \left(t , u\right) = I_{\left[0 , 1\right]} \left(t\right) \frac{3}{8} \left(t^{2} + 2 u\right) + I_{\left(1 , 2\right]} \left(t\right) \frac{9}{14} t^{2} u^{2}$ for $0 \leq u \leq 1$.