The number of times a printing press breaks down each month is Poisson distributed with λ = 4. What is the probability that the printing press breaks down between 2 and 8 times during a particular month?
> A certain gas station sells X thousand gallons of gas each week. Suppose that the cumulative distribution function for X is F (x) = 1 – ¼ (2 - x)2, 0 ≤ x ≤ 2. (a) If the tank contains 1.6 thousand gallons at the beginning of the week, find the probabilit
> Compute the variances of the three random variables whose probability tables are given in Table 6. Relate the sizes of the variances to the spread of the values of the random variable. Table 6:
> Find E(X ), Var(X ), and the standard deviation of X, where X is the random variable whose probability table is given in Table 5. Table 5:
> Table 4 is the probability table for a random variable X. Find E(X ), Var(X ), and the standard deviation of X. Table 4:
> For any positive integer n, the function fn(x) = cn x(n-2)/2 e-x/2, x ≥ 0, where cn is an appropriate constant, is called the chi-square density function with n degrees of freedom. Find c2 and c4 such that f2 (x) and f4 (x) are probability density functi
> For any positive constants k and A, verify that the function f (x) = kAk/xk+1, x ≥ A, is a density function. The associated cumulative distribution function F (x) is called a Pareto distribution. Compute F (x).
> For any number A, verify that f (x) = eA-x, x ≥ A, is a density function. Compute the associated cumulative distribution for X.
> Let X be a continuous random variable on 3 ≤ x ≤ 4, with the density function f (x) = 2(x - 3). (a) Calculate Pr (3.2 ≤ X ) and Pr (3 ≤ X ). (b) Find E(X ) and Var(X ).
> Let X be a continuous random variable on 0 ≤ x ≤ 2, with the density function f (x) = 3/8 x2. (a) Calculate Pr (X ≤ 1) and Pr (1 ≤ X ≤ 1.5). (b) Find E(X ) and Var (X ).
> A pair of dice is rolled until a 7 or an 11 appears, and the number of rolls preceding the final roll is observed. The probability of rolling 7 or 11 is 2/9. What is the probability that at least three consecutive rolls precede the final roll?
> A pair of dice is rolled until a 7 or an 11 appears, and the number of rolls preceding the final roll is observed. The probability of rolling 7 or 11 is 2/9. Determine the average number of consecutive rolls preceding the final roll.
> If the laboratory in Exercise 7 uses batches of 5 instead of 10 samples, the probability of a negative test on the mixture of 5 samples is (.95)5 = .774. Thus, Table 2 gives the probabilities for the number X of tests required. (a) Find E(X ). (b) If the
> A pair of dice is rolled until a 7 or an 11 appears, and the number of rolls preceding the final roll is observed. The probability of rolling 7 or 11 is 2/9. Determine the formula for pn, the probability of exactly n consecutive rolls preceding the final
> A small volume of blood is selected and examined under a microscope, and the number of white blood cells is counted. Suppose that for healthy people the number of white blood cells in such a specimen is Poisson distributed with λ = 4. What is the average
> A small volume of blood is selected and examined under a microscope, and the number of white blood cells is counted. Suppose that for healthy people the number of white blood cells in such a specimen is Poisson distributed with λ = 4. What is the probabi
> A small volume of blood is selected and examined under a microscope, and the number of white blood cells is counted. Suppose that for healthy people the number of white blood cells in such a specimen is Poisson distributed with λ = 4. What is the probabi
> Do the same as in Exercise 29 with a normal random variable. Exercise 29: The Chebyshev inequality says that for any random variable X with expected value m and standard deviation σ, Pr (µ - nσ ≤ X ≤ µ + nσ) ≥ 1 -1/n2. (a) Take n = 2. Apply the Chebyshe
> The Chebyshev inequality says that for any random variable X with expected value m and standard deviation σ, Pr (µ - nσ ≤ X ≤ µ + nσ) ≥ 1 -1/n2. (a) Take n = 2. Apply the Chebyshev inequality to an exponential random variable. (b) By integrating, find th
> (a) Show that about 95% of the area under the standard normal curve lies between -2 and 2. (b) Let X be a normal random variable with expected value µ and variance σ2. Compute Pr (µ - 2σ ≤ X ≤ µ + 2σ).
> It is useful in some applications to know that about 68% of the area under the standard normal curve lies between -1 and 1. (a) Verify this statement. (b) Let X be a normal random variable with expected value µ and variance σ2. Compute Pr (µ - σ ≤ X ≤ µ
> Scores on an entrance exam Scores on a school’s entrance exam are normally distributed, with µ = 500 and σ = 100. If the school wishes to admit only the students in the top 40%, what should be the cutoff grade?
> Let Z be a standard normal random variable. Find the number a such that Pr (a ≤ Z ) = .40.
> A medical laboratory tests many blood samples for a certain disease that occurs in about 5% of the samples. The lab collects samples from 10 persons and mixes together some blood from each sample. If a test on the mixture is positive, an additional 10 te
> Explain how to create a probability density histogram.
> Suppose that the police force in Exercise 23 maintains the same height requirements for women as men and that the heights of women in the city are normally distributed, with µ = 65 inches and σ = 1.6 inches. What percentage of the women are eligible for
> Make a small probability table for a discrete random variable X and use it to define E(X ), Var (X ), and the standard deviation of X.
> What is a discrete random variable?
> What is a probability table?
> What is Pr (X = n) for a geometric random variable with parameter p (the probability of success)? What is E(X ) in this case?
> What is Pr (X = n) for a Poisson random variable with parameter λ? What is E(X ) in this case?
> How is an integral involving a normal density function converted to an integral involving a standard normal density function?
> What is a standard normal random variable? Write the density function.
> What is the density function for a normal random variable with mean µ and standard deviation σ?
> What is the expected value of an exponential random variable?
> What is an exponential density function? Give an example.
> The men hired by a certain city police department must be at least 69 inches tall. If the heights of adult men in the city are normally distributed, with µ = 70 inches and σ = 2 inches, what percentage of the men are tall enough to be eligible for recrui
> Give two ways to compute the variance of a continuous random variable.
> How is the expected value of a continuous random variable computed?
> What is a cumulative distribution function, and how is it related to the corresponding probability density function?
> How is a probability density function used to calculate probabilities?
> What are the two properties of a probability density function?
> What is the difference between a discrete random variable and a continuous random variable?
> Repeat Exercise 2 with λ = .75 and make a histogram. Exercise 2: Let X be a Poisson random variable with parameter λ = 5. Compute the probabilities p0, … , p6 to four decimal places.
> Let X be a Poisson random variable with parameter λ = 5. Compute the probabilities p0, … , p6 to four decimal places.
> Suppose that a random variable X has a Poisson distribution with λ = 3, as in Example 1. Compute the probabilities p6, p7, p8.
> The number of accidents occurring each month at a certain intersection is Poisson distributed with λ = 4.8. (a) During a particular month, are five accidents more likely to occur than four accidents? (b) What is the probability that more than eight acc
> A certain machine part has a nominal length of 80 millimeters, with a tolerance of {.05 millimeter. Suppose that the actual length of the parts supplied is a normal random variable with mean 79.99 millimeters and standard deviation .02 millimeter. How ma
> The number of babies born each day in a certain hospital is Poisson distributed with λ = 6.9. (a) During a particular day, are 7 babies more likely to be born than 6 babies? (b) What is the probability that at most 15 babies will be born during a parti
> The number of people arriving during a 5-minute interval at a supermarket checkout counter is Poisson distributed with λ = 8. (a) What is the probability that exactly eight people arrive during a particular 5-minute period? (b) What is the probability t
> Let X be a Poisson random variable with parameter l. Use Exercise 23 in Section 11.5 to show that the probability that X is an even integer (including 0) is e-λ cosh λ. Exercise 23: The hyperbolic cosine of x, denoted by cosh x, is defined by cosh x = ½
> Let X be a geometric random variable with parameter p. Derive the formula for E(X) by using the power series formula (see Example 3 in Section 11.5): 1 + 2x + 3x2 + … = 1/(1 - x)2 for |x| < 1.
> A person shooting at a target has five successive hits and then a miss. If x is the probability of success on each shot, the probability of having five successive hits followed by a miss is x5(1 - x). Take first and second derivatives to determine the va
> In a production process, a box of fuses is examined and found to contain two defective fuses. Suppose that the probability of having two defective fuses in a box selected at random is (λ2/2) e-λ for some λ. Take first and second derivatives to determine
> Suppose that you toss a fair coin until a head appears and count the number X of consecutive tails that precede it. (a) Determine the probability that exactly n consecutive tails occur. (b) Determine the average number of consecutive tails that occur.
> Suppose that a large number of persons become infected by a particular strain of staphylococcus that is present in food served by a fast-food restaurant and that the germ usually produces a certain symptom in 5% of the persons infected. What is the proba
> Whenever a document is fed into a high-speed copy machine, there is a .5% chance that a paper jam will stop the machine. (a) What is the expected number of documents that can be copied before a paper jam occurs? (b) Determine the probability that at le
> Extensive records are kept of the life spans (in months) of a certain product, and a relative frequency histogram is constructed from the data, using areas to represent relative frequencies (as in Fig. 4 in Section 12.1). It turns out that the upper boun
> Let X be a geometric random variable with parameter p < 1. Find a formula for Pr (X < n), for n > 0. [Note: The partial sum of a geometric series with ratio r is given by 1 + r + … + rn-1 = (1 - rn)/(1 – r). ]
> At a certain junior high school, two-thirds of the students have at least one tooth cavity. A dental survey is made of the students. What is the probability that the first student to have a cavity is the third student examined?
> In a certain town, there are two competing taxicab companies, Red Cab and Blue Cab. The taxis mix with downtown traffic in a random manner. There are three times as many Red taxis as there are Blue taxis. Suppose that you stand on a downtown street corne
> The quality-control department at a sewing machine factory has determined that 1 out of 40 machines does not pass inspection. Let X be the number of machines on an assembly line that pass inspection before a machine is found that fails inspection. (a) W
> Repeat Exercise 10 with p = .6. Exercise 10: If X is a geometric random variable with parameter p = .9, compute the probabilities p0, … , p5 and make a histogram.
> If X is a geometric random variable with parameter p = .9, compute the probabilities p0, … , p5 and make a histogram.
> A bakery makes gourmet cookies. For a batch of 4800 oatmeal and raisin cookies, how many raisins should be used so that the probability of a cookie having no raisins is .01? [Note: A reasonable assumption is that the number of raisins in a random cookie
> During a certain part of the day, an average of five automobiles arrives every minute at the tollgate on a turnpike. Let X be the number of automobiles that arrive in any 1-minute interval selected at random. Let Y be the interarrival time between any tw
> The number of typographical errors per page of a certain newspaper has a Poisson distribution, and there is an average of 1.5 errors per page. (a) What is the probability that a randomly selected page is error-free? (b) What is the probability that a p
> On a typical weekend evening at a local hospital, the number of persons waiting for treatment in the emergency room is Poisson distributed with λ = 6.5. (a) What is the likelihood that either no one or only one person is waiting for treatment? (b) What
> A piece of new equipment has a useful life of X thousand hours, where X is a random variable with the density function f (x) = .01xe-x/10, x ≥ 0. A manufacturer expects the machine to generate $5000 of additional income for every thousand hours of use, b
> The monthly number of fire insurance claims filed with the Firebug Insurance Company is Poisson distributed with λ = 10. (a) What is the probability that in a given month no claims are filed? (b) What is the probability that in a given month no more th
> Repeat Exercise 2 with λ = 2.5 and make a histogram. Exercise 2: Let X be a Poisson random variable with parameter λ = 5. Compute the probabilities p0, … , p6 to four decimal places.
> In a large factory there is an average of two accidents per day, and the time between accidents has an exponential density function with an expected value of ½ day. Find the probability that the time between accidents will be less than 8 hours (1/3 day).
> In a large factory there is an average of two accidents per day, and the time between accidents has an exponential density function with an expected value of ½ day. Find the probability that the time between two accidents will be more than ½ day and less
> Find (by inspection) the expected values and variances of the exponential random variables with the density function given. 1.5e-1.5x
> Find (by inspection) the expected values and variances of the exponential random variables with the density function given. .2e-0.2x
> Find (by inspection) the expected values and variances of the exponential random variables with the density function given. ¼ e-x/4
> Find (by inspection) the expected values and variances of the exponential random variables with the density function given. 3e-3x
> Use the integral routine to convince yourself that ∫-∞∞ x2 f (x) dx = 1, where f (x) is the standard normal density function. [Note: Since f (x) approaches zero so rapidly as x gets large in magnitude, the value of the improper integral is nearly the sam
> The computations of the expected value and variance of an exponential random variable relied on the fact that, for any positive number k, be-kb and b2 e-kb approach 0 as b gets large. That is, lim x→∞ x ekx = 0 and lim
> An exponential random variable X has been used to model the relief times (in minutes) of arthritic patients who have taken an analgesic for their pain. Suppose that the density function for X is f (x) = k e-kx and that a certain analgesic provides relief
> If the lifetime (in weeks) of a certain brand of lightbulb has an exponential density function and 80% of all lightbulbs fail within the first 100 weeks, find the average lifetime of a lightbulb.
> Recall that the median of an exponential density function is that number M such that Pr (X ≤ M) = ½. Show that M = (ln 2)/k. (We see that the median is less than the mean.)
> Let X be the time to failure (in years) of a computer chip, and suppose that the chip has been operating properly for a years. Then, it can be shown that the probability that the chip will fail within the next b years is Pr (a ≤ X ≤ a + b) / Pr (a ≤ X).
> The Math SAT scores of a recent freshman class at a university were normally distributed, with µ = 535 and σ = 100. (a) What percentage of the scores were between 500 and 600? (b) Find the minimum score needed to be in the top 10% of the class.
> A certain type of bolt must fit through a 20-millimeter test hole or else it is discarded. If the diameters of the bolts are normally distributed, with µ = 18.2 millimeters and σ = .8 millimeters, what percentage of the bolts will be discarded?
> Which route should the student in Exercise 29 take if she leaves home at 7:26 a.m.? Exercise 29: A student with an eight o’clock class at the University of Maryland commutes to school by car. She has discovered that along each of two possible routes her
> A student with an eight o’clock class at the University of Maryland commutes to school by car. She has discovered that along each of two possible routes her traveling time to school (including the time to get to class) is approximately a normal random va
> The amount of weight required to break a certain brand of twine has a normal density function, with µ = 43 kilograms and σ = 1.5 kilograms. Find the probability that the breaking weight of a piece of the twine is less than 40 kilograms.
> If the amount of milk in a gallon container is a normal random variable, with µ = 128.2 ounces and σ = .2 ounce, find the probability that a random container of milk contains less than 128 ounces.
> Suppose that the life span of a certain automobile tire is normally distributed, with µ = 25,000 miles and σ = 2000 miles. (a) Find the probability that a tire will last between 28,000 and 30,000 miles. (b) Find the probability that a tire will last mo
> The condenser motor in an air conditioner costs $300 to replace, but a home air-conditioning service will guarantee to replace it free when it burns out if you will pay an annual insurance premium of $25. The life span of the motor is an exponential rand
> The gestation period (length of pregnancy) of a certain species is approximately normally distributed with a mean of 6 months and standard deviation of 12 month. (a) Find the percentage of births that occur after a gestation period of between 6 and 7 mo
> Calculate the area under the standard normal curve for values of z (a) between .5 and 1.5, (b) between -.75 and .75, (c) to the left of -.3, (d) to the right of -1.
> Let Z be a standard normal random variable. Calculate (a) Pr (-1.3 ≤ Z ≤ 0) (b) Pr (.25 ≤ Z) (c) Pr (-1 ≤ Z ≤ 2.5) (d) Pr (Z ≤ 2)
> Show that the function f (x) = e-(1/2)[(x-µ)/σ]2 has inflection points at x = m ± s.
> Show that the function f (x) = e-x2/2 has inflection points at x = ± 1.
> Show that the function f (x) = e-(1/2)[(x-m)/s]2 has a relative maximum at x = m.
> Show that the function f (x) = e-x2/2 has a relative maximum at x = 0.
> Find the expected values and the standard deviations (by inspection) of the normal random variables with the density function given. 1/5√2π e-(1/2)[(x-3)/5]2
> Find the expected values and the standard deviations (by inspection) of the normal random variables with the density function given. 1/3√2π e-(1/18)x2
> Find the expected values and the standard deviations (by inspection) of the normal random variables with the density function given. 1/√2π e-(1/2)(x+5)2
