In a certain town the length of residence of a family in a

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.

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?

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.

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?

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?

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?

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?

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.

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 point selected randomly. What are the chances that the two resulting pieces and the piece of length a can be used to form a triangular pen? Hint: Three segments form a triangle if and only if the length of any one of them is less than the sum of the lengths of the remaining two.

Let be a random number between /2 and /2. Find the probability density function of X = tan .

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.

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

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.

Let X be a continuous random variable with distribution function F. Prove that F (X) is uniformly distributed over (0, 1).

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.

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).

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.

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

Let X N (, 2). Find the probability distribution function of |X | and its expected value.

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

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?

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?

Let (,) and Z N (0, 1); find E(eZ).

Let X N (0, 2). Calculate the density function of Y = X2.

Let X N (, 2). Calculate the density function of Y = eX.

Let X N (0, 1). Calculate the density function of Y = |X|.

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 out of this book? For simplicity, assume that a month is 30 days.

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?

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?

Prove that for some constant k, f (x) = kax2 , a (0,), is a normal probability density function.

(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.

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.

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 selected bottle?

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 selected bottle?

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

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.