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

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

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

[

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

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

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

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

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

$$\text{ Find }$$

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

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

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

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

Let X and Y denote the lengths of life, in years, of two components in an electronic system. If the joint density function of these variables is $f(x,y)=e^{-(x+y)}$, for $x \gt 0$, $y \gt 0$, $f(x,y)=0$, elsewhere, find $P(0 \lt X \lt 1 | Y = 2).$

Let X denote the diameter of an armored electric cable and Y denote the diameter of the ceramic mold that makes the cable. Both X and Y are scaled so that they range between 0 and 1. Suppose that X and Y have the joint density

$$f(x,y) = \frac{1}{y}, \text{ for } 0\leq x,y\leq q, \quad f(x,y)=0 \text{ elsewhere}$$

Find $P(X+Y \text{ greater than }\dfrac{1}{2})$.

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.

$$\text{Suppose that } X \text{ and } Y \text{ have the following joint probability distribution: }$$

$$\begin{matrix} & & x & \\ \hline & & 2 & 4 \\ & 1 & 0.10 & 0.15 \\ y & 3 & 0.20 & 0.30 \\ & 5 & 0.10 & 0.15 \end{matrix}$$

[

$$(a) \text{ Find the marginal distribution of } X.$$

$$(b) \text{ Find the marginal distribution of } Y.$$

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

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

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

[

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

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

The amount of kerosene, in thousands of liters, in a tank at the beginning of any day is a random amount Y from which a random amount X is sold during that day. Suppose that the tank is not resupplied during the day so that x $\leq$ y, and assume that the joint density function of these variables is f(x,y)=2, for 0$\lt x \leq y \lt 1$, f(x,y)=0, elsewhere. (a) Determine if X and Y are independent. (b) Find $P(1/4\lt X \lt 1/2 | Y=3/4).$

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

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

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

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

The joint density function for random variables X and Y is f(x,y) = {C(x+y) if 0≤x≤3, 0≤y≤2 , 0 otherwise. Find P(X+Y≤1)

$$\text{ Let } X, Y , \text{ and } Z \text{ have the joint probability density function }$$

$$f(x,y,z)=kxy^2z, \text{ for } 0<x,y<1, \text{ } 0<z<2,$$

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

[

$$(a) \text{ Find } k.$$

$$(b) \text{ Find } P(X<\frac{1}{4}, Y>\frac{1}{2}, 1<Z<2).$$

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.

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

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

find

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

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

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

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

A coin is tossed twice. Let Z denote the number of heads on the first toss and W the total number of heads on the 2 tosses. If the coin is unbalanced and a head has a 40% chance of occurring, find (a) the joint probability distribution of W and Z; (b) the marginal distribution of W; (c) the marginal distribution of Z; (d) the probability that at least 1 head occurs.