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

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

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

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

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

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

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

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

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

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

$$\text{ Find }$$

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

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

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

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

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.

An emergency braking parachute system on a military aircraft consists of a large parachute of diameter 6 m. If the airplane mass is 8500 kg, and it lands at 400 km/hr, find the time and distance at which the airplane is slowed to 100 km/hr by the parachute alone. Plot the aircraft speed versus distance and versus time. What is the maximum “g-force” experienced? An engineer proposes that less space would be taken up by replacing the large parachute with three noninterfering parachutes each of diameter 3.75 m. What effect would this have on the time and distance to slow to 100 km/hr?

The temperature T (in degrees Fahrenheit) of a patient t hours after arriving at the emergency room of a hospital at 10:00 P.M. is given by $T(t)=98.6+4 \cos \frac{\pi t}{36}, \quad 0 \leq t \leq 18$. Find the patient’s temperature at (a) 10:00 P.M., (b) 4:00 A.M., and (c) 10:00 A.M. (d) At what time do you expect the patient’s temperature to return to normal? Explain your reasoning.

The following exercises describe apportionment problems for situations besides the House of Representatives. Do the following: a. Find the apportionment by Hamilton's method. b. Find the apportionment by Jefferson's method. c. Find the apportionment by Webster's method. d. Find the apportionment by the Hill-Huntington method. e. Compare the results of the various methods. A city plans to purchase 16 new emergency vehicles. They are to be apportioned among 485 members of the police department, 213 members of the fire department, and 306 members of the paramedic squad.

You are driving a car when a deer suddenly darts across the road in front of you. Your brain registers the emergency and sends a signal to your foot to hit the brake. The car travels a reaction distance D, in feet, during this time, where D is a function of the speed r, in miles per hour, that the car is traveling when you see the deer, given by $D(r)=\frac{11 r+5}{10}.$ Find the inverse and explain what it represents. Is the inverse a function?

A group of hikers is lost in a national park. Two ranger stations have received an emergency SOS signal from the hikers. Station B is 75 miles due east of station A. The bearing from station A to the signal is S 60° E and the bearing from station B to the signal is S 75° W. A rescue party is in the park 20 miles from station A at a bearing of S 80° E. Find the distance and the bearing the rescue party must travel to reach the lost hikers.

B.L., 17, had a compression fracture at the C5-C6 after jumping into a river from a bridge and slamming into a submerged rock. Fortunately, she was saved by a companion who had first-aid expertise and worked to limit any collateral damage. B.L. was unable to move her limbs, feel touch, or have reflexes in her limbs when she arrived at the emergency room. Describe why it is impossible to determine the entire amount of lasting damage during this initial time frame.

You have an emergency, and you ask your employer for a loan. She gives you $\$ 100$. You sign a form indicating that you received the loan and will start repaying it next month. However, you quit the job the following week. Your former manager calls and asks whether you are going to pay back the loan. You indicate that since you are a minor, the contract is voidable, meaning that you do not have to abide by it. Is it a good idea to avoid the contract?

A patient is evaluated in the emergency department after causing an automobile accident while being under the influence of alcohol. While assessing the patient, which statement would be the most therapeutic?

1. “Why did you drive after you had been drinking?”
2. “We have multiple patients to see tonight as a result of this accident.”
3. “Tell me what happened before, during, and after the automobile accident tonight.”
4. “It will be okay. No one was seriously hurt in the accident.”

A nuclear reactor is equipped with two independent automatic shutdown systems to shut down the reactor when the core temperature reaches the danger level. Neither system is perfect. System A shuts down the reactor 90% of the time when the danger level is reached. System B does so 80% of the time. The reactor is shut down if either system works.

Now simulate 100 trials of the reactor's response to an emergency of this kind. Estimate the probability that it will shut down. This probability is higher than the probability that either system working alone will shut down the reactor.

Study of why EMS workers leave the job. A study of fulltime emergency medical service (EMS) workers published in the Journal of Allied Health (Fall 2011) found that only about 3% leave their job in order to retire. (See Exercise $3.45$, p. 158.) Assume that the true proportion of all fulltime EMS workers who leave their job in order to retire is $p=.03$. In a random sample of 1,000 full-time EMS workers, let $\hat{p}$ represent the proportion who leave their job in order to retire. Compute $P(\hat{p}<.05)$. Interpret this result.