A straight rod is formed by connecting three sections A,

Find the 0.5, 0.25, 0.75, 0.1, and 0.9 quantiles of the standard normal distribution.

Suppose that X has the normal distribution for which the mean is 1 and the variance is 4. Find the value of each of the following probabilities: a. Pr(X 3) b. Pr(X > 1.5) c. Pr(X = 1) d. Pr(2

If the temperature in degrees Fahrenheit at a certain location is normally distributed with a mean of 68 degrees and a standard deviation of 4 degrees, what is the distribution of the temperature in degrees Celsius at the same location?

Find the 0.25 and 0.75 quantiles of the Fahrenheit temperature at the location mentioned in Exercise 3.

Let X1, X2, and X3 be independent lifetimes of memory chips. Suppose that each Xi has the normal distribution with mean 300 hours and standard deviation 10 hours. Compute the probability that at least one of the three chips lasts at least 290 hours.

. If the m.g.f. of a random variable X is (t) = et2 for

Suppose that the measured voltage in a certain electric circuit has the normal distribution with mean 120 and standard deviation 2. If three independent measurements of the voltage are made, what is the probability that all three measurements will lie between 116 and 118?

Evaluate the integral  0 e3x2 dx

A straight rod is formed by connecting three sections A, B, and C, each of which is manufactured on a different machine. The length of section A, in inches, has the normal distribution with mean 20 and variance 0.04. The length of sectionB, in inches, has the normal distribution with mean 14 and variance 0.01. The length of sectionC, in inches, has the normal distribution with mean 26 and variance 0.04. As indicated in Fig. 5.6, the three sections are joined so that there is an overlap of 2 inches at each connection. Suppose that the rod can be used in the construction of an airplane wing if its total length in inches is between 55.7 and 56.3. What is the probability that the rod can be used?

. If a random sample of 25 observations is taken from the normal distribution with mean and standard deviation 2, what is the probability that the sample mean will lie within one unit of

Suppose that a random sample of size n is to be taken from the normal distribution with mean and standard deviation 2. Determine the smallest value of n such that Pr(|Xn | < 0.1) 0.9.

a. Sketch the c.d.f.  of the standard normal distribution from the values given in the table at the end of this book. b. From the sketch given in part (a) of this exercise, sketch the c.d.f. of the normal distribution for which the mean is 2 and the standard deviation is 3.

Suppose that the diameters of the bolts in a large box follow a normal distribution with a mean of 2 centimeters and a standard deviation of 0.03 centimeter. Also, suppose that the diameters of the holes in the nuts in another large box follow the normal distribution with a mean of 2.02 centimeters and a standard deviation of 0.04 centimeter. A bolt and a nut will fit together if the diameter of the hole in the nut is greater than the diameter of the bolt and the difference between these diameters is not greater than 0.05 centimeter. If a bolt and a nut are selected at random, what is the probability that they will fit together?

Suppose that on a certain examination in advanced mathematics, students from university A achieve scores that are normally distributed with a mean of 625 and a variance of 100, and students from university B achieve scores which are normally distributed with a mean of 600 and a variance of 150. If two students from university A and three students from university B take this examination, what is the probability that the average of the scores of the two students from university A will be greater than the average of the scores of the three students from university B? Hint: Determine the distribution of the difference between the two averages.

Suppose that 10 percent of the people in a certain population have the eye disease glaucoma. For persons who have glaucoma, measurements of eye pressure X will be normally distributed with a mean of 25 and a variance of 1. For persons who do not have glaucoma, the pressure X will be normally distributed with a mean of 20 and a variance of 1. Suppose that a person is selected at random from the population and her eye pressure X is measured. a. Determine the conditional probability that the person has glaucoma given that X = x. b. For what values of x is the conditional probability in part (a) greater than 1/2?

Suppose that the joint p.d.f. of two random variables X and Y is f (x, y) = 1 2 e(1/2)(x2+y2) for

Consider a random variable X having the lognormal distribution with parameters and 2. Determine the p.d.f. of X.

Suppose that the random variables X and Y are independent and that each has the standard normal distribution. Show that the quotient X/Y has the Cauchy distribution

Suppose that the measurement X of pressure made by a device in a particular system has the normal distribution with mean and variance 1, where is the true pressure. Suppose that the true pressure is unknown but has the uniform distribution on the interval [5, 15]. If X = 8 is observed, find the conditional p.d.f. of given X = 8.

Let X have the lognormal distribution with parameters 3 and 1.44. Find the probability that X 6.05.

Let X and Y be independent random variables such that log(X) has the normal distribution with mean 1.6 and variance 4.5 and log(Y ) has the normal distribution with mean 3 and variance 6. Find the distribution of the product XY .

Suppose that X has the lognormal distribution with parameters and 2. Find the distribution of 1/X.

Suppose that X has the lognormal distribution with parameters 4.1 and 8. Find the distribution of 3X1/2.

The method of completing the square is used several times in this text. It is a useful method for combining several quadratic and linear polynomials into a perfect square plus a constant. Prove the following identity, which is one general form of completing the square: n i=1 ai(x bi) 2 + cx =  n i=1 ai x n i=1 aibi c/2 n i=1 ai 2 +n i=1 ai  bi n i=1 aibi n i=1 ai 2 +  n i=1 ai 1  c n i=1 aibi c2/4  if n i=1 ai = 0.

In Example 5.6.10, we considered the effect of continuous compounding of interest. Suppose that S0 dollars earn a rate of r per year componded continuously for u years. Prove that the principal plus interest at the end of this time equals S0eru. Hint: Suppose that interest is compounded n times at intervals of u/n years each. At the end of each of the n intervals, the principal gets multiplied by 1 + ru/n. Take the limit of the result as n .

Let X have the normal distribution whose p.d.f. is given by (5.6.6). Instead of using the m.g.f., derive the variance of X using integration by p