Let X denote the number of times a certain numerical control machine will malfunction: 1, 2, or 3 times on any given day. Let Y denote the number of times a technician is called on an emergency call. Their joint probability distribution is given as

$$\begin{matrix} & & & x & \\ \hline & & 1 & 2 & 3 \\ & 1 & 0.05 & 0.05 & 0.10 \\ y & 3 & 0.05 & 0.10 & 0.35 \\ & 5 & 0.00 & 0.20 & 0.10 \end{matrix}$$

(a) Evaluate the marginal distribution of X. (b) Evaluate the marginal distribution of Y. (c) Find P(Y=3|X=2).

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) $P(X \lt Y)$.

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 $P(1/4\lt X \lt 1/2 | Y=3/4).$

A candy company distributes boxes of chocolates with a mixture of creams, toffees, and cordials. Suppose that the weight of each box is 1 kilogram, but the individual weights of the creams, toffees, and cordials vary from box to box. For a randomly selected box, let X and Y represent the weights of the creams and the toffee, respectively, and suppose that the joint density function of these variables is f(x,y) = 24xy, for 0 $\leq x \leq 1,$0 $\leq y \leq 1, x+ y \leq 1,$ f(x,y)=0, elsewhere. (a) Find the probability that in a given box the cordials account for more than 1/2 of the weight. (b) Find the marginal density for the weight of the creams. (c) Find the probability that the weight of the toffees in a box is less than 1/8 of a kilogram if it is known that creams constitute 3/4 of the weight.

Three cards are drawn without replacement from the 12 face cards (jacks, queens, and kings) of an ordinary deck of 52 playing cards. Let X be the number of kings selected and Y the number of jacks. Find (a) the joint probability distribution of X and Y; (b) P[(X, Y) in A, where A is the region given by ${(x, y) | x + y \geq 2 }.$

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

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

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

[

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

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

Determine the values of c so that the following functions represent joint probability distributions of the random variables X and Y: (a) f(x,y)=cxy, for x=1, 2, 3; y=1, 2, 3; (b) f(x,y)=c|x-y|, for x=-2, 0, 2; y=-2,3.

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 ${(x,y) | x+y \leq 2 }.$

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 $P(0 \lt X \lt 1 | Y = 2).$

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

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

find

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

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

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

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

The length of time, in minutes, for an airplane to obtain clearance for takeoff at a certain airport is a random variable Y=3X−2, where X has the density function $f(x)=\frac{1}{4}e^{-x/4}$, for $x \gt 0$, f(x)=0, elsewhere. Find the mean and variance of the random variable Y.

From a sack of fruit containing 3 oranges, 2 apples, and 3 bananas, a random sample of 4pieces of fruit is selected. If is the number of oranges and is the number of apples in the sample, find

$$\text{(a) } f(y|2) \text{ for all values of } y;$$

$$\text{(b) } P(Y=0 | X=2).$$

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

Each rear tire on an experimental airplane is supposed to be filled to a pressure of 40 pounds per square inch (psi). Let X denote the actual air pressure for the right tire and Y denote the actual air pressure for the left tire. Suppose that X and Y are random variables with the joint density function $f(x,y) = k(x^2+y^2)$, for $30 \leq x \leq 50$, $30 \leq y \leq 50$, f(x,y)=0, elsewhere. (a) Find k. (b) Find $P(30 \leq X \leq 40$ and $40 \leq Y \lt 50)$. (c) Find the probability that both tires are underfilled.