Fundamentals of Probability, with Stochastic Processes 3rd Edition solutions

Fundamentals of Probability, with Stochastic Processes 3
It takes a professor a random time between 20 and 27 minutes to walk from his home to school every day. If he has a class at 9:00 A.M. and he leaves home at 8:37 A.M., find the probability that he reaches his class on time.

Fundamentals of Probability, with Stochastic Processes 3
Suppose that 15 points are selected at random and independently from the interval (0, 1). How many of them can be expected to be greater than 3/4?

Fundamentals of Probability, with Stochastic Processes 3
The time at which a bus arrives at a station is uniform over an interval (a, b) with mean 2:00 P.M. and standard deviation 12 minutes. Determine the values of a and b.

Fundamentals of Probability, with Stochastic Processes 3
Suppose that b is a random number from the interval (3, 3). What is the probability that the quadratic equation x2 + bx + 1 = 0 has at least one real root?

Fundamentals of Probability, with Stochastic Processes 3
The radius of a sphere is a random number between 2 and 4. What is the expected value of its volume? What is the probability that its volume is at most 36?

Fundamentals of Probability, with Stochastic Processes 3
A point is selected at random on a line segment of length $. What is the probability that the longer segment is at least twice as long as the shorter segment?

Fundamentals of Probability, with Stochastic Processes 3
A point is selected at random on a line segment of length $. What is the probability that none of the two segments is smaller than $/3?

Fundamentals of Probability, with Stochastic Processes 3
From the class of all triangles one is selected at random. What is the probability that it is obtuse? Hint: The largest angle of a triangle is less than 180 degrees but greater than or equal to 60 degrees.

Fundamentals of Probability, with Stochastic Processes 3
A farmer who has two pieces of lumber of lengths a and b (a < b) decides to build a pen in the shape of a triangle for his chickens. He sends his foolish son out to cut the longer piece and the boy, without taking any thought as to the ultimate purpose, makes a cut on the lumber of length b, at a poi

Fundamentals of Probability, with Stochastic Processes 3
Let be a random number between /2 and /2. Find the probability density function of X = tan .

Fundamentals of Probability, with Stochastic Processes 3
Let X be a random number from [0, 1]. Find the probability mass function of [nX], the greatest integer less than or equal to nX.

Fundamentals of Probability, with Stochastic Processes 3
Let X be a random number from (0, 1). Find the density functions of (a) Y = ln(1 X), and (b) Z = Xn.

Fundamentals of Probability, with Stochastic Processes 3
Let X be a uniform random variable over the interval (0, 1+), where 0 < < 1 is a given parameter. Find a function of X, say g(X), so that E 4 g(X)5 = 2.

Fundamentals of Probability, with Stochastic Processes 3
Let X be a continuous random variable with distribution function F. Prove that F (X) is uniformly distributed over (0, 1).

Fundamentals of Probability, with Stochastic Processes 3
Let g be a nonnegative real-valued function on R that satisfies the relation # g(t) dt = 1. Show that if, for a random variable X, the random variable Y = # X g(t) dt is uniform, then g is the density function of X.

Fundamentals of Probability, with Stochastic Processes 3
The sample space of an experiment is S = (0, 1), and for every subset A of S, P (A) = # A dx. Let X be a random variable defined on S by X() = 5 1. Prove that X is a uniform random variable over the interval (1, 4).

Fundamentals of Probability, with Stochastic Processes 3
Let Y be a random number from (0, 1). Let X be the second digit of Y . Prove that for n = 0, 1, 2, . . . , 9, P (X = n)increases as n increases. This is remarkable because it shows that P (X = n), n = 1, 2, 3,... is not constant. That is, Y is uniform but X is not.

Fundamentals of Probability, with Stochastic Processes 3
Find the expected value and the variance of a random variable with the probability density function f (x) = F 2 e2(x1)2 .

Fundamentals of Probability, with Stochastic Processes 3
Let X N (, 2). Find the probability distribution function of |X | and its expected value.

Fundamentals of Probability, with Stochastic Processes 3
Determine the value(s) of k for which the following is the probability density function of a normal random variable. f (x) = kek2x22kx1 , < x < .

Fundamentals of Probability, with Stochastic Processes 3
The viscosity of a brand of motor oil is normal with mean 37 and standard deviation 10. What is the lowest possible viscosity for a specimen that has viscosity higher than at least 90% of that brand of motor oil?

Fundamentals of Probability, with Stochastic Processes 3
In a certain town the length of residence of a family in a home is normal with mean 80 months and variance 900. What is the probability that of 12 independent families, living on a certain street of that town, at least three will have lived there more than eight years?

Fundamentals of Probability, with Stochastic Processes 3
Let (,) and Z N (0, 1); find E(eZ).

Fundamentals of Probability, with Stochastic Processes 3
Let X N (0, 2). Calculate the density function of Y = X2.

Fundamentals of Probability, with Stochastic Processes 3
Let X N (, 2). Calculate the density function of Y = eX.

Fundamentals of Probability, with Stochastic Processes 3
Let X N (0, 1). Calculate the density function of Y = |X|.

Fundamentals of Probability, with Stochastic Processes 3
Suppose that the odds are 1 to 5000 in favor of a customer of a particular bookstore buying a certain fiction bestseller . If 800 customers enter the store every day, how many copies of that bestseller should the store stock every month so that, with a probability of more than 98%, it does not run ou

Fundamentals of Probability, with Stochastic Processes 3
Every day a factory produces 5000 light bulbs, of which 2500 are type I and 2500 are type II. If a sample of 40 light bulbs is selected at random to be examined for defects, what is the approximate probability that this sample contains at least 18 light bulbs of each type?

Fundamentals of Probability, with Stochastic Processes 3
To examine the accuracy of an algorithm that selects random numbers from the set {1, 2, . . . , 40}, 100,000 numbers are selected and there are 3500 ones. Given that the expected number of ones is 2500, is it fair to say that this algorithm is not accurate?

Fundamentals of Probability, with Stochastic Processes 3
Prove that for some constant k, f (x) = kax2 , a (0,), is a normal probability density function.

Fundamentals of Probability, with Stochastic Processes 3
(a) Prove that for all x > 0, 1 x 2 * 1 1 x2 , ex2/2 < 1 -(x) < 1 x 2 ex2/2 . Hint: Integrate the following inequalities: (1 3y4 )ey2/2 < ey2/2 < (1 + y2 )ey2/2 . (b) Use part (a) to prove that 1 -(x) 1 x 2 ex2/2 . That is, as x , the ratio of the two sides approaches 1.

Fundamentals of Probability, with Stochastic Processes 3
Let Z be a standard normal random variable. Show that for x > 0, lim t P * Z > t + x t | Z t , = ex . Hint: Use part (b) of Exercise 31.

Fundamentals of Probability, with Stochastic Processes 3
The amount of soft drink in a bottle is a normal random variable. Suppose that in 7% of the bottles containing this soft drink there are less than 15.5 ounces, and in 10% of them there are more than 16.3 ounces. What are the mean and standard deviation of the amount of soft drink in a randomly select

Fundamentals of Probability, with Stochastic Processes 3
The amount of soft drink in a bottle is a normal random variable. Suppose that in 7% of the bottles containing this soft drink there are less than 15.5 ounces, and in 10% of them there are more than 16.3 ounces. What are the mean and standard deviation of the amount of soft drink in a randomly select

Fundamentals of Probability, with Stochastic Processes 3
In a forest, the number of trees that grow in a region of area R has a Poisson distribution with mean R, where is a positive real number. Find the expected value of the distance from a certain tree to its nearest neighbor

Fundamentals of Probability, with Stochastic Processes 3
Let I = # 0 ex2/2 dx; then I 2 = E 0 8 E 0 e(x2+y2)/2 dy9 dx. Let y/x = s and change the order of integration to show that I 2 = /2. This gives an alternative proof of the fact that - is a distribution function. The advantage of this method is that it avoids polar coordinates.