UNIVERSIDAD DE LA SALLE
Departamento de Ciencias B´
asicas
´
Area de Matem´
aticas y Estad´ıstica
Estad´ıstica 1Algunas distribuciones de probabilidad discretas y continuas
Binomial distribution, Geometric distribution, Negative Binomial Distribution, Hyper-geometric distribution and Poisson distribution.
1. For each scenario described below, state whether or not the binomial distribution is a reasonable model for the random variable and why. State any assumptions you make.
a) A production process produces thousands of temperature transducers. Let X denote the number of nonconforming transducers in a sample of size 30 selected at random from the process.
b) Four identical electronic components are wired to a controller that can switch from a failed component to one of the remaining spares. Let X denote the number of components that have failed after a specified period of operation.
c) Defects occur randomly over the surface of a semiconductor chip. However, only 80 % of defects can be found by testing. A sample of 40 chips with one defect each is tested. LetX
denote the number of chips in which the test finds a defect.
d) LetX denote the number of surface flaws in a large coil of galvanized steel.
2. The random variableX has a binomial distribution withn= 10 andp= 0,5. Sketch the proba-bility mass function of X.
a) Sketch the probability mass function ofX.
1) What value ofX is most likely? 2) What value(s) ofX is(are) least likely?
b) Determine the following probabilities:
1) P(X = 5) 2) P(X ≤2) 3) P(X ≥9) 4) P(3≤X <5)
3. The phone lines to an airline reservation system are occupied 40 % of the time. Assume that the events that the lines are occupied on successive calls are independent. Assume that 10 calls are placed to the airline.
a) What is the probability that for exactly three calls the lines are occupied?
b) What is the probability that for at least one call the lines are not occupied?
c) What is the expected number of calls in which the lines are all occupied?
4. A multiple choice test contains 25 questions, each with four answers. Assume a student just guesses on each question.
a) What is the probability that the student answers more than 20 questions correctly?
5. This exercise illustrates that poor quality can affect schedules and costs. A manufacturing process has 100 customer orders to fill. Each order requires one component part that is purchased from a supplier. However, typically, 2 % of the components are identified as defective, and the components can be assumed to be independent.
a) If the manufacturer stocks 100 components, what is the probability that the 100 orders can be filled without reordering components?
b) If the manufacturer stocks 102 components, what is the probability that the 100 orders can be filled without reordering components?
c) If the manufacturer stocks 102 components, what is the probability that the 100 orders can be filled without reordering components?
6. Suppose the random variableX has a geometric distribution withp= 0,5. Determine the following probabilities:
a) P(X = 1)
b) P(X = 4)
c) P(X ≤2)
d) P(X ≥2)
7. Suppose the random variableX has a geometric distribution with a mean of 2,5. Determine the following probabilities:
a) P(X = 1)
b) P(X = 4)
c) P(X ≤3)
d) P(X >3)
8. Suppose thatX is a negative binomial random variable withp= 0,2 andr= 4. Determine the following:
a) E[X]
b) P(X = 20)
c) P(X = 19)
d) P(X = 21)
e) The most likely value forX.
9. In a clinical study, volunteers are tested for a gene that has been found to increase the risk for a disease. The probability that a person carries the gene is 0.1.
a) What is the probability 4 or more people will have to be tested before 2 with the gene are detected?
b) How many people are expected to be tested before 2 with the gene are detected?
10. Assume that each of your calls to a popular radio station has a probability of 0.02 of connecting, that is, of not obtaining a busy signal. Assume that your calls are independent.
a) What is the probability that your first call that connects is your tenth call?
b) What is the probability that it requires more than five calls for you to connect?
c) What is the mean number of calls needed to connect?
12. A fault-tolerant system that processes transactions for a financial services firm uses three separate computers. If the operating computer fails, one of the two spares can be immediately switched online. After the second computer fails, the last computer can be immediately switched online. Assume that the probability of a failure during any transaction is and that the transactions can be considered to be independent events.
a) What is the mean number of transactions before all computers have failed?
b) What is the variance of the number of transactions before all computers have failed?
13. Suppose X has a hypergeometric distribution with N= 100,n= 4, andK= 20. Determine the following:
a) P(X = 1)
b) P(X = 6)
c) P(X = 4)
d) Determine the mean and variance ofX.
14. Suppose X has a hypergeometric distribution with N = 10, n = 3, and K = 4. Sketch the probability mass function of X.
15. Determine the cumulative distribution function forX in Exercise 14.
16. A company employs 800 men under the age of 55. Suppose that 30 % carry a marker on the male chromosome that indicates an increased risk for high blood pressure.
a) If 10 men in the company are tested for the marker in this chromosome, what is the proba-bility that exactly 1 man has the marker?
b) If 10 men in the company are tested for the marker in this chromosome, what is the proba-bility that more than 1 has the marker?
17. A state runs a lottery in which 6 numbers are randomly selected from 40, without replacement. A player chooses 6 numbers before the state’s sample is selected.
a) What is the probability that the 6 numbers chosen by a player match all 6 numbers in the state’s sample?
b) What is the probability that 5 of the 6 numbers chosen by a player appear in the state’s sample?
c) What is the probability that 4 of the 6 numbers chosen by a player appear in the state’s sample?
d) If a player enters one lottery each week, what is the expected number of weeks until a player matches all 6 numbers in the state’s sample?
18. Suppose X has a Poisson distribution with a mean of 4. Determine the following probabilities:
a) P(X = 0)
b) P(X ≤2)
c) P(X = 4)
19. Suppose that the number of customers that enter a bank in an hour is a Poisson random variable, and suppose thatP(X = 0) = 0,05. Determine the mean and variance of X.
20. The number of telephone calls that arrive at a phone exchange is often modeled as a Poisson random variable. Assume that on the average there are 10 calls per hour.
b) What is the probability that there are 3 or less calls in one hour?
c) What is the probability that there are exactly 15 calls in two hours?
d) What is the probability that there are exactly 5 calls in 30 minutes?
21. The number of cracks in a section of interstate highway that are significant enough to require repair is assumed to follow a Poisson distribution with a mean of two cracks per mile.
a) What is the probability that there are no cracks that require repair in 5 miles of highway?
b) What is the probability that at least one crack requires repair in mile of highway?
c) If the number of cracks is related to the vehicle load on the highway and some sections of the highway have a heavy load of vehicles whereas other sections carry a light load, how do you feel about the assumption of a Poisson distribution for the number of cracks that require repair?
22. The number of failures of a testing instrument from contamination particles on the product is a Poisson random variable with a mean of 0.02 failure per hour.
a) What is the probability that the instrument does not fail in an 8-hour shift?
b) What is the probability of at least one failure in a 24-hour day?
Continuous Uniform distribution, Normal distribution and Exponential distribution.
23. Suppose X has a continuous uniform distribution over the interval [ 1,5,5,5 ]
a) Determine the mean, variance, and standard deviation ofX.
b) What isP(X <2,5)?
24. The net weight in pounds of a packaged chemical herbicide is uniform for 49,75 < x < 50,25 pounds.
a) Determine the mean and variance of the weight of packages.
b) Determine the cumulative distribution function of the weight of packages.
c) DetermineP(X <50,1)
25. The probability density function of the time it takes a hematology cell counter to complete a test on a blood sample isf(x) = 0,2 for 50< x <75 seconds.
a) What percentage of tests require more than 70 seconds to complete.
b) What percentage of tests require less than one minute to complete.
c) Determine the mean and variance of the time to complete a test on a sample.
26. Determine the following probabilities for the standard normal random variableZ:
a) P(Z <2)
b) P(Z <1,32)
c) P(Z >1,45)
d) P(Z >−2,15)
e) P(−2< Z <2)
27. AssumeZ has a standard normal distribution. Determine the value forzthat solves each of the following:
a) P(Z < z) = 0,9
c) P(Z > z) = 0,1
d) P(−1,24< Z < z) = 0,8
e) P(−z < Z < z) = 0,95
28. AssumeX is normally distributed with a mean of 5 and a standard deviation of 4. Determine the following:
a) P(X <11)
b) P(X >0)
c) P(3< X <7)
d) P(−2< X <9)
e) P(2< X <8)
29. Assume X is normally distributed with a mean of 10 and a standard deviation of 2. Determine the value forxthat solves each of the following:
a) P(X > x) = 0,5
b) P(X > x) = 0,95
c) P(x < X <10) = 0,2
d) P(−x < X−10< x) = 0,95
e) P(−x < X < x) = 0,99
30. The time it takes a cell to divide (called mitosis) is normally distributed with an average time of one hour and a standard deviation of 5 minutes.
a) What is the probability that a cell divides in less than 45 minutes?
b) What is the probability that it takes a cell more than 65 minutes to divide?
c) What is the time that it takes approximately 99 % of all cells to complete mitosis?
31. The length of an injection-molded plastic case that holds magnetic tape is normally distributed with a length of 90.2 millimeters and a standard deviation of 0.1 millimeter.
a) What is the probability that a part is longer than 90.3 millimeters or shorter than 89.7 millimeters?
b) What should the process mean be set at to obtain the greatest number of parts between 89.7 and 90.3 millimeters?
c) If parts that are not between 89.7 and 90.3 millimeters are scrapped, what is the yield for the process mean that you selected in part (b)?
32. The sick-leave time of employees in a firm in a month is normally distributed with a mean of 100 hours and a standard deviation of 20 hours.
a) What is the probability that the sick-leave time for next month will be between 50 and 80 hours?
b) How much time should be budgeted for sick leave if the budgeted amount should be exceeded with a probability of only 10 %?
33. The weight of a sophisticated running shoe is normally distributed with a mean of 12 ounces and a standard deviation of 0.5 ounce.
a) What is the probability that a shoe weighs more than 13 ounces?
c) If the standard deviation remains at 0.5 ounce, what must the mean weight be in order for the company to state that 99,9 % of its shoes are less than 13 ounces?
34. Suppose X has an exponential distribution withλ= 2. Determine the following:
a) P(X ≤0)
b) P(X ≥2)
c) P(X ≤1)
d) P(1< X <2)
e) Find the value of x such thatP(X < x) = 0,05.
35. Suppose the counts recorded by a geiger counter follow a Poisson process with an average of two counts per minute.
a) What is the probability that there are no counts in a 30-second interval?
b) What is the probability that the first count occurs in less than 10 seconds?
c) What is the probability that the first count occurs between 1 and 2 minutes after start-up?
36. The time to failure (in hours) for a laser in a cytometry machine is modeled by an exponential distribution with
a) What is the probability that the laser will last at least 20,000 hours?
b) What is the probability that the laser will last at most 30,000 hours?
c) What is the probability that the laser will last between 20,000 and 30,000 hours?
37. The time between calls to a plumbing supply business is exponentially distributed with a mean time between calls of 15 minutes.
a) What is the probability that there are no calls within a 30-minute interval?
b) What is the probability that at least one call arrives within a 10-minute interval?
c) What is the probability that the first call arrives within 5 and 10 minutes after opening?
d) Determine the length of an interval of time such that the probability of at least one call in the interval is 0.90.
Nota:Ejercicios tomados de Applied Scientists and Probability for Engineers, Third Edition. Montgomery D., Runger G. John Wiley & Sons 2003.
