$$\text{ The joint probability density function of the random variables } X, Y , \text{ and } Z \text{ is }$$

$$f(x,y,z)=\frac{4xyz^2}{9}, \text{ for } 0<x,y<1, \text{ } 0<z<3,$$

$$f(x,y,z)=0, \text{ elsewhere.}$$

$$\text{ Find }$$

$$(a) \text{ the joint marginal density function of } Y \text{ and } Z;$$

$$(b) \text{ the marginal density of } Y ;$$

$$(c) P( \frac{1}{4} < X < \frac{1}{2}, Y > \frac{1}{3} , 1 < Z < 2);$$

$$(d) P(0 < X < \frac{1}{2} | Y = \frac{1}{4}, Z = 2).$$

Suppose the random variables X, Y, and Z have the joint probability density function f(x, y, z) = 8xyz for 0 < x < 1, 0 < y < 1, and 0 < z < 1. Determine the following: (a) Conditional probability distribution of X given that Y = 0.5 and Z = 0.8 (b) P(X < 0.5 | Y = 0.5, Z = 0.8)

Suppose that the random variables X, Y, and Z have the joint probability density function f(x, y, z) = 8xyz for 0 < x < 1, 0 < y < 1, and 0 < z < 1. Determine the following: a. P(X < 0.5) b. (X < 0.5, Y < 0.5) c. P( Z < 2) d. P(X < 0.5 or Z < 2) e. E(X) f. P(X < 0.5 | Y = 0.5) g. P(X < 0.5, Y < 0.5 | Z = 0.8) h. Conditional probability distribution of X given that Y = 0.5 and Z = 0.8 i. P(X < 0.5 | Y = 0.5, Z = 0.8)

$$\text{ Given the joint density function }$$

$$f(x,y) = \frac{6-x-y}{8}, \text{ for } 0<x<2, \text{ } 2<y<4,$$

$$f(x,y)=0, \text{ elsewhere,}$$

$$\text{ find } P(1<Y<3 |X=1).$$

A service facility operates with two service lines. On a randomly selected day, let X be the proportion of time that the first line is in use whereas Y is the proportion of time that the second line is in use. Suppose that the joint probability density function for (X, Y) is $f(x,y)=\frac{3}{2}(x^2+y^2)$, for $0 \leq x,y \leq 1$, f(x,y)=0, elsewhere. (a) Compute the probability that neither line is busy more than half the time.(b) Find the probability that the first line is busy more than 75% of the time.

$$\text{ The joint density function of the random variables } X \text{ and } Y \text{ is }$$

$$f(x,y)=6x, \text{ for } 0<x<1, \text{ } 0<y<1-x,$$

$$f(x,y)=0, \text{ elsewhere.}$$

[

$$(a) \text{ Show that } X \text{ and } Y \text{ are not independent. }$$

$$(b) \text{ Find } P(X >0.3 | Y = 0.5).$$

Let X denote the reaction time, in seconds, to a certain stimulus and } Y denote the temperature ( in degrees Fahrenheit at which a certain reaction starts to take place. Suppose that two random variables X and Y have the joint density f(x,y)=4xy, for $0 \lt x \lt 1$, $0 \lt y \lt 1$, f(x,y)=0, elsewhere. Find (a) $P(0 \leq X \leq \frac{1}{2} \text{ and } \frac{1}{4} \leq Y \leq \frac{1}{2});$ (b) $P(X \lt Y)$.

Let the joint probability density function of random variables X and Y be bivariate normal. Show that if $\sigma_X$ = $\sigma_Y$ , then X + Y and X − Y are independent random variables.

Hint: Show that the joint probability density function of X + Y and X − Y is bivariate normal with correlation coefficient 0.

$$\text{ If the joint probability distribution of } X \text{ and } Y \text{ is given by }$$

$$f(x,y)=\frac{x+y}{30}, \text{ for } x=0,1,2,3; \text{ } y=0,1,2,$$

find

$$\text{(a) } P(X \leq 2, Y=1);$$

$$\text{(b) } P(X>2,Y \leq 1);$$

$$\text{(c) } P(X>Y);$$

$$\text{(d) } P(X+Y=4).$$

Let X, Y, Z have joint pdf f(x, y, z) = 2(x + y + z)/3, 0 < x < 1, 0 < y < 1, 0 < z < 1, zero elsewhere.

(a) Find the marginal probability density functions of X, Y, and Z.

(b) Compute $P \left( 0 < X < \frac { 1 } { 2 } , 0 < Y < \frac { 1 } { 2 } , 0 < Z < \frac { 1 } { 2 } \right)$ and $P \left( 0 < X < \frac { 1 } { 2 } \right) = P ( 0 < Y < \frac { 1 } { 2 } ) = P \left( 0 < Z < \frac { 1 } { 2 } \right)$

(c) Are X, Y , and Z independent?

(d) Calculate $E \left( X^{ 2 } Y Z + 3 X Y^{ 4 } Z^{ 2 } \right)$

(e) Determine the cdf of X, Y, and Z.

(f) Find the conditional distribution of X and Y, given Z = z, and evaluate E(X + Y |z).

(g) Determine the conditional distribution of X, given Y = y and Z = z, and compute E(X|y, z).

A fast-food restaurant operates both a drive-through facility and a walk-in facility. On a randomly selected day, let $X$ and $Y$, respectively, be the proportions of the time that the drive-through and walk-in facilities are in use, and suppose that the joint density function of these random variables is

$f(x,y)=\frac{2}{3}(x+2y), \text{ ~~for~~ } 0 \leq x \leq 1$, $0 \leq y \leq 1,$

$f(x,y)=0$, elsewhere.

(a) Find the marginal density of $X$.

(b) Find the marginal density of $Y$.

(c) Find the probability that the drive-through facility is busy less than one-half of the time.

From a sack of fruit containing 3 oranges, 2 apples, and 3 bananas, a random sample of 4 pieces of fruit is selected. If $X$ is the number of oranges and $Y$ is the number of apples in the sample, find (a) the joint probability distribution of $X$ and $Y$; (b) $P[(X,Y) \in A]$, where A is the region that is given by ${(x,y) | x+y \leq 2 }.$

$$\text{ Consider the random variables } X \text{ and } Y \text{ with joint density function }$$

$$f(x,y)=x+y, \text{ for } 0 \leq x,y \leq 1,$$

$$f(x,y)=0, \text{ elsewhere.}$$

[

$$(a) \text{ Find the marginal distributions of } X \text{ and } Y .$$

$$(b) \text{ Find } P(X >0.5, Y > 0.5).$$

$$\text{ If the joint density function of } X \text{ and } Y \text{ is given by }$$

$$f(x,y)=\frac{2}{7}(x+2y), \text{for } 0<x<1, \text{ } 1<y<2,$$

$$f(x,y)=0, \text{ elsewhere,}$$

$$\text{ find the expected value of } g(X,Y)= \frac{X}{Y^3} + X^2Y.$$