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Introduction To Probability 1st Edition Mark Daniel Ward, Ellen Gundlach - Solutions

Let X have density fx(x) = 2ln x /x , for 1 ≤ x ≤ e and fx(x) = 0 otherwise. Find Var(X).
Find the variance of the given random variable. When meeting a friend at a coffee house, Donald sees that his friend has not yet arrived, so he assumes that his friend's arrival time, X, in minutes, will have density fx(x) = 1/5 for 0 ≤ x ≤ 5 and fx(x) = 0 otherwise
Every day, a student calls his mother and then (afterwards) calls his girlfriend. Let X be the time (in hours) until he calls his mother, and let Y be the time (in hours) until he calls his girlfriend. Since he always calls his mother first, then X < Y. So let the joint density of the time
Let X have densityfx(x) = 1/3e-x/3,for 0 < x,and fx(x) = 0 otherwise.a. Find the expected value of X3, i.e., E(X3).b. Find a general form, which handles all positive integers n, for E(Xn), often called the nth moment of X.
Show that, if X1,..., Xn are independent random variables, and a1,..., an are constants, then
If the joint density of X and Y is 8/27xy on the interior of the triangle with corners at (0, 0), (3, 0), (0, 3) (and the joint density is 0 elsewhere), find E(X).
Find the variance of the given random variable. The waiting time X, in minutes, until the next green car passes has density fx(x) = l/2-e-x/2, for 0 < x, and fx(x) = 0 otherwise.
Find the variance of the given random variable. Darius always arrives at Wiley dining hall 50 minutes before it closes. He estimates that the time X necessary to stand in line, in minutes, has density fx(x) = x/50, = for 0 ≤ x ≤ 10, and fx(x) = 0 otherwise.
Find the variance of the given random variable. Let X have density fx(x) = 1/4, for 2 ≤ x ≤ 6, and fx(x) = 0 otherwise.
Find the variance of the given random variable. Upon arriving at a restaurant and finding that no table is available, Diana and Markus have a waiting time X (in hours) with density Fx(x) = 2e-2x, for 0 < x, and fx(x) = 0 otherwise.
Find the variance of the given random variable. Let X be the time, in hours, that Christine spends talking on the telephone. Then X has density fx(x) = 3e-3x, for 0 < x, and fx(x) = 0 otherwise.
Consider a pair of random variables X, Y with constant joint density on the triangle with vertices at (0, 0), (3, 0), and (0, 3).a. Find the expected value of the sum of X and Y, i.e., find E(X + Y).b. Find the variance of X, i.e., find Var (X).
Using the following probability density functiona. Find the cumulative distribution function Fx(x).b. Find P(0.25 c. Find P{X > 0.45).d. Find P(X = 0.25).e. Find the mean E(X).f. Find the standard deviation Ïƒxg. Graph the density fx(x).h. Graph the CDF Fx(x). Identify the median on the
Which of the following can be true? If an answer is false, state why it is falsea. The area under the Fx(x) curve from - ∞ to ∞ is 1.b. The area under the fx(x) curve from - ∞ to ∞ is 1.
Which of the following can be true? If an answer is false, state why it is false a. The CDF Fx(x) can be negative. b. The density fx(x) can be negative. c. The mass px(x) can be negative.
Which of the following can be true? If an answer is false, state why it is falsea. The graph of fx(x) can have jumps (i.e., can be discontinuous).b. The graph of Fx(x) for discrete random variables can have jumps.c. The graph of Fx(x) for continuous random variables can have jumps.d. The graph of
Using the following probability density function,a. Find the cumulative distribution function Fx{x).b. Find P(X > 4).c. Find P(-2 ‰¤ X ‰¤ 12).d. Find P(X e. Find the mean K(X).f. Find the standard deviation ax-g. Graph the density fx(x).h. Graph the CDF Fx(x). Identify the median
Using the following probability density function,a. Find the cumulative distribution function Fx{x).b. Find P{X > 3).c. Find P(-0.5 ≤ X < 5).d. Find P(X < 4.2).e. Find the mean K(X).
Using the following CDF,Find the corresponding density fx(x).
What is the constant k that makes the following function a valid density?
What is the constant k that makes the following function valid density?
Which of the following can be true? If an answer is false, state why it is false a. A CDF Fx(x) can have the value 4.3. b. A density fx(x) can have the value 4.3. c. A mass px(x) can have the value 4.3.
Which of the following can be true? If an answer is false, state why it is false a. A CDF Fx(x) can be 1 for two or more values of x. b. A density fx(x) can be 1 for two or more values of x. c. A mass px(x) can be 1 for two or more values of x.
Which of the following can be true? If an answer is false, state why it is false a. The (indefinite) integral of Fx(x) is fx(x). b. The (indefinite) integral of fx(x) is Fx(x). c. The area under the curve of fx(x) between - ∞ and x is Fx(x). d. The area under the curve of Fx(x) between - ∞ and
In a certain manufacturing process, an automated quality control computer checks 10 yards of rope at a time. If no defects are detected in that 10-yard section, that portion of the rope is passed on. However, if there is a defect detected, a person will have to check the rope over more carefully to
Consider the two-dimensional house in Figure. Suppose that a fly is found somewhere in the house, with a location that is uniformly distributed.a. Find the conditional density of Y given that X = 2.b. Find the conditional probability that the fly is upstairs, i.e., Y > 10, given that X = 2, i.e.,
Consider the two-dimensional house in Figure. Suppose that a fly is still found somewhere in the house, with joint densityfX,Y(x,y) = 1/150for x,y in the house,and fX,Y(x,y) = 0 otherwise. (We no longer assume X = 2.)a. Find E(X).b. Find Var(X).c. Find E(Y).d. Find Var(y).
Suppose that a particular long jumper assumes that each of his jumps is uniformly distributed between 6.5 and 7.2 meters. He is happy whenever he jumps 7 meters or more. If he makes 10 such jumps, what is the probability that he is happy with exactly 4 of his jumps?
A machine manufactures cubes with a side length which varies uniformly over the interval [0.2, 0.3] in millimeters. For the following problems, make sure you use the correct units. (Assume the sides of the base and the height is all the same.)a. What is the expected side length?b. What is the
Kelly throws a dart at a circular dartboard of radius 3 feet. Let X and Y denote the location where the dart lands. Assume that -3 ≤ X ≤ 3 and -3 ≤ Y ≤ 3 and X2 + Y2 ≤ 9, i.e., the dart lands on the dartboard. Moreover, assume that the dart's location is Uniform on the dartboard,
An automatic camera is taking pictures of insects that land on a wire. The distance between each insect and the camera (at the time of the picture) is random, with constant density on the interval [1, 6], measured in feet. (Assume that the locations of the insects are independent.) A 'Very good"
For the scenario in Exercise 31.16, if X,Y,Z are the distances of the camera from the insect for his first, second, and third pictures, respectively, then find the expected value of the smallest of these three distances, i.e., find E(min(X, Y, Z)).
You are doing two loads of laundry: one was just put in the washer; the other was simultaneously put in the dryer. You know it takes the dryer between 25 and 40 minutes to completely dry your clothes, and the washer takes between 27 and 37 minutes to complete its cycle. Those distributions are
In a game of chance, a circle of radius 3 inches is drawn on a piece of paper that is 8.5 x 11 sq. in. While blindfolded, the student tries to place her pencil tip inside the circle. She wins $3 if she is successful, or loses $1 if unsuccessful. (If she misses the paper altogether, she can try
You are driving down an interstate when you suddenly realize you have missed your exit because you were busy listening to an exciting chapter of a Harry Potter book on CD. According to the CD case, the chapter lasts for 14 minutes. Assume you are driving 60 miles per hour (in other words, 1 mile
Emmanuel makes pots out of clay. The clay costs $5.00 per pound. Each pot is made from a random amount X of clay (in pounds), uniformly distributed between 1.9 and 2.7 pounds. There is also a charge to bake the pot in a kiln (a kiln is a big oven for making clay gets very hard). The charge to use
A child is playing in a sandbox and loses a very small toy. If the sand box is a square the measures 5 feet by 5 feet, what is the probability that the toy is actually located in a right triangle at the southwest corner of the sandbox, with sides measuring 1.5 feet by 2.5 feet?
The location of a mole rat is Uniform inside a circular enclosure that has diameter 40 feet. What is the probability that I he mole rat is within 2 feet from the edge of the enclosure?
Assume a mother's child has a random location that is uniformly distributed across the 80 foot by 120 foot playground shown in Figure below.a. What is the probability that the child is in the grass?b. On the swings?c. On the slides?
In a crayon factory, wax is rolled into cylinders; each of which are exactly 3.6 inches long, but the radius (in inches) is a Uniform random variable X on the interval [0.15, 0.17]. Find the probability that the volume of a crayon exceeds 0.30 cubic inches.
Let X, Y, Z be independent and uniformly distributed on the interval [0, 10]. Find the probability that Y is the middle value, i.e., find P(X < Y < Z or Z < Y < X).
Suppose X, Y have constant joint density on the triangle with vertices at (0, 0), (3, 0), and (0, 3). Find E(X).
There is a fly randomly located uniformly in a room that is 10 feet high, 14 feet long, and 13 feet wide. What is the probability that the fly is within 1 foot of the walls, ceiling, or floor?
Let X1, X2, X3 be independent continuous random variables, each uniformly distributed in the interval [0, 10]. Let Y denote the middle of the three values. Find the cumulative distribution function FY(a) = P(Y ≤ a) of the random variable Y.
If X has a Continuous Uniform distribution on the interval (1, 10), find E (lnX).
You believe that there is a fly somewhere less than 6 feet away from you. If you believe that he is located uniformly in a circle of radius 6 feet away from you, what is the probability that he is more than 2 feet away from you?
A quarterback needs to throw the ball quickly, and in his haste, the location it lands is uniformly distributed within a 30 x 120 sq. ft. area in the end zone. The receiver can only catch the ball the ball lands within 7 feet of him, i.e., within a circle of radius 7 feet, completely contained in
A wall in a room is 108 inches tall and 132 inches wide. There is a painting on the wall that is 18 inches by 24 inches. If a tennis ball is accidentally flung at the wall, and the location where it lands is uniformly distributed on the wall, what is the probability that the tennis ball hits the
A solar cell measures 10 inches by 10 inches, and it has 4 non-overlapping regions that generate electricity. Each of these regions is 4 inches by 4 inches in size (there is a border between the electricity-generating regions). What is the probability that a photon that hits the solar cell at a
Chickens at Rolling Meadows Farm lay an average of 18 eggs per day. The farmer has rigged a fancy monitoring device to the nesting boxes so that he can monitor exactly when the hens lay their eggs. Assume that no 2 eggs will be laid at exactly the same time and that the eggs (and chickens) are
Let X be uniform on [0, 10]. Let Y be exponential with E (Y) = 5. Find P (X < Y).
Let X be exponential with expected value 3. Let Y be another random variable that depends on X as follows: if X > 5, then Y = X - 5; otherwise, Y = 0. a. Find the expected value of Y. b. Find the variance of Y.
The time between consecutive uses of a vending machine is Exponential with average 15 minutes. a. Given that the machine has not been used in the previous 5 minutes, what is the probability that the machine will not be used during the next 10 minutes? b. How many purchases are expected within the
Dan, Dominic, and Doug are waiting together in the living room for their girlfriends, Sally. Shellie, and Susanne, to call their waiting times (in hours) are independent Exponential random variables, with parameters 2.1, 3.7, and 5.5, respectively. What is the probability that the phone will ring
Air traffic control stations often have insufficient numbers of air traffic controllers, sometimes just one person on duty. In a recent study, a lone air traffic controller is managing an airstrip in which the expected time between arrivals of airplanes is 15 minutes. Assume that the times between
Consider an Exponential random variable X with parameter λ > 0. Let Y = [X], which means we get Y by rounding X down to the barest integer (in particular, Y itself is a discrete random variable, because Y is always an integer). For example, if X = 7.2, then Y = 7. If X = 12.9999, then Y = 12. If X
Let X be an Exponential random variable with E(X) = 1/λ. Define Y = [X], i.e., Y is the least integer that is greater than or equal to Y. For instance, if X = 5.3, then Y = 6. If X = 4.99, then Y = 5. If X = 5.00001 then Y = 6. If X = 5.99999 then Y = 6. What is the distribution of Y?
A store manager is very impatient at the start of the day. Let X be the time (in minutes) until his first customer arrives. The workers at the store decided to find a way to measure his impatience level as a function of X:g(X) = aX2 + bX + c,Where a, b, c are fixed constants.a. Calculate the
On the average, hurricane of category 4 or stronger (on the Stafford/Simpson scale) strikes the United States once every 6 years. A hurricane of this strength has winds of at least 131 miles per hour and can cause extreme damage, (www.aoml.noaa. gov/hrd/Landsea/deadly/index,html) An insurance
Suppose that the time in between customers at a store are independent Exponential random variables, with an average of 2 minutes between consecutive customers. Let X be the time until the 3rd customer arrives.a. Find E(X).b. Find Var(X).c. Find the density of X
Bob is waiting for his girlfriend Alice to call. His waiting time X is exponentially distributed, with expected waiting time K(X) = 0.20 hours, i.e., 12 minutes. a. What is the probability that he must wait more than 12 minutes for her to call? b. What is the probability that he must wait more than
A student waits for a bus. Let X be the number of hours that the student waits. Assume that the waiting time is Exponential with average 20 minutes. a. What is the probability that the student waits more than 30 minutes? b. What is the probability that the student waits more than 45 minutes
It is your birthday and you are waiting for someone to write a "Happy Birthday" message on your Face book wall. Your waiting lime is approximately Exponential with average waiting time of 10 minutes between such postings; assume that the times of the postings are independent. a. What is the
Lily estimates that her time to fall asleep each night is approximately Exponential, with an average time of 30 minutes until she falls asleep. a. What is the probability that it takes her less than 10 minutes to fall asleep? b. What is the probability that it takes her more than 1 hour to fall
Suppose that, when an airplane waits on the runway, the company must pay each customer a fee if the waiting time exceeds 3 hours. Suppose that an airplane with 72 passengers waits an exponential amount of time on the runway, with average 1.5 hours. If the waiting time X, in hours, is bigger than 3,
Chickens at Rolling Meadows Farm lay an average of 18 eggs per day. The farmer has rigged a fancy monitoring device to the nesting boxes so that he can monitor exactly when the hens lay their eggs. Assume that no 2 eggs will be laid at exactly the same time and that the eggs (and chickens) are
While listening to "Pinball Wizard," you decide to conduct an experiment in which you play pinball over and over again. Each game takes an Exponential amount of time to finish, with expected value of 3 minutes.a. What is the density of the time that you spend, if you play two games in a row?b. What
A student procrastinates, watching cars pass his house. After 10 black cars have passed, he will (finally) start his homework. He notices that the time between consecutive cars that pass is Exponentially distributed, and the times are independent, each with expected time of 1/5 of a minute. He also
If X is a Gamma random variable with r = 2, show that X does not have the memory less property.
On the average, a category 4 (on the Stafford/Simpson scale) or stronger hurricane strikes the United States once every 6 years. A hurricane of this strength has winds of at least 131 miles per hour and can cause extreme damage, (www.aoml.noaa.gov/ hrd/Landsea/deadly/index.html) An insurance agency
The waiting time for rides at an amusement park has an Exponential distribution with average waiting time of 1/2 an hour (assume that the waiting times are independent). a. If a person rides 5 rides, what is the expected amount of time that the person spends waiting in line? b. If a person rides 5
The proportion of people who pass a professional qualifying exam on the first try has a Beta distribution with α = 3 and β = 4. a. What is the expected proportion of people who will pass on the first try at the next exam? b. What is the standard deviation in the proportion of people who will pass
A grocery store chain is trying to decide how much shelf space to devote to organic produce. If they don't stock enough, their more upscale customers will shop at the competing grocery store. If they stock too much, then much of the expensive organic produce will have to be thrown out when it
The proportion of horses with a particular coloration is modeled by the following function:a. Why is this Beta distribution?b. What are the parameters?c. What is the value of k?d. What is the expected value?
Email checking. The percent of time on a workday that employees at a company spend checking email can be modeled by the following function:a. Why is this Beta distribution?b. What are the parameters?c. What is the value of k?d. What is the expected value?
Use the Normal table to find the following probabilities starting from Z. Also sketch a standard Normal curve, and shade the region corresponding to the given probability. a. P (Z < 1.47) b. P (Z > 1.47) c. P (Z < -1.47) d. P (Z > -1.47) e. P (Z = -1.47)
The students in my class have Exam 1 scores which are normally distributed with a mean of 75 and a standard deviation of 9. If a student is selected at random,a. What is the probability the student will have a score of more than 90 (an A)?b. What is the probability the student will have a score of
The quantity of sugar X (measured in grams) in a randomly selected piece of candy is normally distributed, with expected value E(X) = μx = 22 and variance Var(X) = σ2X = 8. Find the probability that a randomly selected piece of candy has less than 20 grams of sugar.
Assume that the annual precipitation in a student's hometown is normally distributed, with expected value μx = 36.3 inches and variance σ2X = 8.41. A rare species of frog lives in the town. This rare species of frog is known to reproduce during the year only if the annual precipitation is between
The distance a student lives (in miles) from their probability classroom is approximately normally distributed with a mean of 3 miles and a standard deviation of 1.2 miles.a. How far away do the closest 10% of students live?b. What is the probability that a student will live too close to get a
Children's movies run an average of 98 minutes with a standard deviation of 10 minutes. You check out a movie, selected at random without reading the running time on the box, from the library to entertain your kids so you can study for your probability test. Assume that your kids will be occupied
A full-grown sunflower stands 12 feet tall, on average, with standard deviation of 1 foot. Find the probability that a sunflower is between 11 to 13 feet tall.
Use the Normal table to find the following probabilities starting from Z. Also sketch a standard Normal curve, and shade the region corresponding to the given probability. a. P (Z ≤ 0.19) b. P (Z ≤ 1.90) c. P (Z ≥ 9.10) d. P (0.19 < Z < 1.90) e. P (Z ≤ -1.90) f. P (Z = -1.90)
Chocolate-coated Sugar Fun Blast cereal is filled into boxes by weight, with the weight approximately normally distributed with an average of 16 ounces and a standard deviation of 0.2 ounces.a. What is the probability that the box you buy (chosen at random) will weigh less than 15.5 ounces?b. What
Assume that the height of an American female is Normal with expected value μ = 64 and standard deviation σ = 2.5.a. What is the probability that an American female's height is 66 inches or taller?b. The heights of 10 American females are measured (in inches). Let Y be the number of the 10 females
Assume that X has a Normal distribution with a mean of 2 and a standard deviation of 3. Use the Normal table to find the following probabilities starting from X. Also sketch a standard Normal curve, and shade the region corresponding to the given probability. a. P (X < 1.62) b. P (X > -8.49) c. P
Assume that X has a Normal distribution with a mean of -1.32 and a standard deviation of 0.34. Use the Normal table to find the following probabilities starting from X. Also sketch a standard Normal curve, and shade the region corresponding to the given probability.a. P (X > -2)b. P (X <
Use the standard Normal table to find the following cut-off values for Z. Also sketch a standard Normal curve, and shade the region corresponding to the given probability. a. P (Z z) = 0.15 c. P (-z < Z < z) = 0.65
Use the standard Normal table to find the following cut-off values for Z. Also sketch a standard Normal curve, and shade the region corresponding to the given probability. a. P (Z < z) = 0.5 b. P (Z > z) = 0.87 c. P (-z < Z < z) = 0.25
Assume that X has a Normal distribution with a mean of 2 and a standard deviation of 3. Use the standard Normal table to find the following cut-off values for X. Also sketch a standard Normal curve, and shade the region corresponding to the given probability.a. P (X < x) = 0.78b. P (X > x) =
Assume that X has a Normal distribution with a mean of -1.32 and a standard deviation of 0.34. Use the standard Normal table to find the following cut-off values for X. Also sketch a standard Normal curve, and shade the region corresponding to the given probability.a. What is the cut-off for the
Find the value of a so that, if Z is a standard Normal random variable, then P(a ≤ Z ≤ 0.54) = 0.3898.
The time that it takes a random person to get a haircut is normally distributed, with an average of 23.8 minutes and a standard deviation of 5 minutes. Assume that different people have independent times of getting their hair cut. Find the probability that, if there are four customers I in a row
Preparing for their dates on a Friday night, it takes each girl (approximately) a normally distributed amount of time to prepare, with an average of 40 minutes (per girl) to get ready, and a standard deviation of 15 minutes. The 50 men who are waiting for them in the lobby are mostly science
After planting 20 evergreen tree saplings on his farm, Don wonders about their heights. If the heights are independent and normally distributed, with average height 3 feet, and standard deviation of 1.2 feet, what is the probability that none of the 20 trees' heights exceeds 5 feet?
On a busy interstate highway, I there are 21 cars in a particular lane. Let X1... X20 denote the 20 distances between these 21 cars (i.e., X1 is the distance between the first and second cars; X2 is the distance between the second and third cars; etc.). At a particular moment, the Xj's are judged
An athletic director is recording long-jump scores for a group of students. The expected value of each of their long jumps is 7 meters, and the standard deviation is 0.2 meters. If 20 such jumps are independent and approximately normally distributed, find the probability that I the average of these
Roberto and Sally are busy making cakes for a bake sale. They need to make 35 cakes for the sale. The cooking times of the cakes are independent and normally distributed. Sally makes 19 of the cakes I and Roberto makes 16 of them. Sally's have average cook time of 45 minutes each, with standard
As in Exercise 35.12, assume that the annual precipitation in a student's hometown is normally distributed, with expected value μ = 36.3 inches and variance σ2 = 8.41. Also assume that the amount of precipitation is independent in distinct years. Let X1... X10 denote the precipitation in the 10
A printer can produce, on average, 30 pages per minute, i.e., one page every 2 seconds. Each page's printing time has standard deviation of 0.3 seconds. If the pages run times are independent, find the probability that the total print time for a 30 page job is 62 seconds or less.

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