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) $P(X \lt Y)$.

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.

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

$$\text{ Determine whether the two random variables } X \text{ and } Y \text{ are dependent or independent, if they are given in the following way. }$$

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$$\text{ Let } X \text{ denote the reaction time, in seconds, to a certain stimulus and } Y \text{ denote the temperature ( in degrees Fahrenheit ) at which a certain reaction starts to take place. Suppose that two random variables } X \text{ and } Y \text{ have the joint density }$$

$$f(x,y) = 4xy, \text{ for } 0<x<1, \text{ } 0<y<1,$$

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

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.

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

A driver’s reaction time to visual stimulus is normally distributed with a mean of 0.4 seconds and a standard deviation of 0.05 seconds. a. What is the probability that a reaction requires more than 0.5 seconds? b. What is the probability that a reaction requires between 0.4 and 0.5 seconds? c. What reaction time is exceeded 90% 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

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

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

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

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

In the senior year of a high school graduating class of 100 students, 42 studied mathematics, 68 studied psychology, 54 studied history, 22 studied both mathematics and history, 25 studied both mathematics and psychology, 7 studied history but neither mathematics nor psychology, 10 studied all three subjects, and 8 did not take any of the three. Randomly select a student from the class and find the probabilities of the following events. (a) A person enrolled in psychology takes all three subjects. (b) A person not taking psychology is taking both history and mathematics.

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

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

Find the covariance of the random variables X and Y.

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

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

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

$$\text{ Find }$$

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

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

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

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

A chemical system that results from a chemical reaction has two important components among others in a blend. The joint distribution describing the proportions $X_1$ and $X_2$ of these two components is given by $f(x_1,x_2)=6x_2$, for $0 \lt x_2 \lt x_1 \lt 1,$ $f(x_1,x_2)=0$, elsewhere. (a) Give the marginal distribution $f_{X_1} (x_1)$ of the proportion} $X_1$ and verify that it is a valid density function. (b) What is the probability that proportion $X_2$ is less than 0.5, given that X1 is 0.7?

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. Find the covariance of the random variables X and Y.