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.

Determine whether the two random variables, given in the following way, are dependent or independent. 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

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

Max needs

$$10 \frac { 1 } { 4 }$$

cups of flour to make a batch of pizza dough for the pizzeria. He only has

$$4 \frac { 1 } { 2 }$$

cups of flour. How much more flour does he need to make the dough'?

Determine the truth value for each statement when p is false, q is true, and r is false.

$$\sim p \rightarrow q$$

Each front tire of a vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable -X for the right tire and Y for the left tire, with joint pdf

$$f(x, y)=\left\{\begin{array}{cl} K\left(x^2+y^2\right) & 20 \leq x \leq 30, \quad 20 \leq y \leq 30 \\ 0 & \text { otherwise } \end{array}\right.$$

a. What is the value of $K$ ? b. What is the probability that both tires are underfilled? c. What is the probability that the difference in air pressure between the two tires is at most 2 psi? d. Determine the (marginal) distribution of air pressure in the right tire alone. e. Are $X$ and $Y$ independent rv's?

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.

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) $P(X \lt Y)$.

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, ad suppose that the joint density function of these random variables is $f(x,y)=\frac{2}{3}(x+2y)$, for $0 \leq x \leq 1$, $0 \leq y \leq 1$, f(x,y)=0, elsewhere.} Find the covariance of the random variables X and Y.

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.summation n=1 to infinity 10^n/(-9)^n-1

Solve the equation graphically. Then check your solution algebraically. 10x - 18x = 4x - 6

A patient with an infection has not responded appreciably to antibiotic therapy, and the nurse suspects antibiotic resistance. What phenomenon is known to contribute to acquired antibiotic resistance?

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

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

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

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

[

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

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

$$\text{ Random variables } X \text{ and } Y \text{ follow a joint distribution }$$

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

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

$$\text{ Determine the correlation coefficient between } X \text{ and } Y.$$