Exam Week Office Hours
Monday, May 11
2 - 3 p.m.
Tuesday, May 12
4 - 5 p.m.
Wednesday, May 13
11- 12 a.m. and 2:30 - 3:30 p.m.
Thursday, May
14, 2 - 3 p.m.
Friday,
May 15, 11 - 12 a.m.
Note: Corrections are marked in boldface.
FINAL EXAM INFORMATION AND REVIEW SUGGESTIONS (Includes more problems and some hints, answers, and solutions that were not passed out in class.)
Exam will be Friday, May 15, 2 - 5 p.m. in CPE 2.216 (Please note: There are two M362kKsections with final exams at the same time; be sure you have the correct room.) Exam will be comprehensive
Note: You will need to bring a non-programmable calculator and a
copy of the appendix (pp. 388 - 389) to the exam with you.
Studying Suggestions
1. Go over your midsemester exams and the solutions to them to be sure you understand the points you missed on the exams.
2. Go over the reviews for the midsemester exams.
3. Do any studying you need to do on material covered since the last exam.
4. Then try yourself out on the following problems, and do more studying if needed .
Review Problems (First installment)
(Remember the usual caveat: these problems are not intended to tell you just what will be on the exam.)
Caution: Formatting and special symbols have been lost in putting this on theweb. Youcan get a hard copy of the second installment and the hints, etc. outsidethe door of RLM 10.136.
1. Two dice are rolled. Let E be the event that the sum of the outcomes is odd and F be the event that at least one of the outcomes is 1. Describe in words each of the events E«F, EC«F, EC«FC, and (E«F)C.
2. A card is drawn at random from an ordinary deck of 52 cards. What
is the probability that it is
a. a black ace or a red queen
b. a face or a black card
c. neither a heart nor a queen?
3. In a certain metropolitan area, 25% of the crimes occur during the day and 80% of the crimes occur in the city. If only 10% of the crimes occur outside the city during the day, what percent occur inside the city during the night? What percent occur outside the city during the night?
4. How many six-digit positive whole numbers are there? How many of them have at least one digit equal to 5?
5. Experience has shown that every new textbook published by a certain publishing house captures randomly between 4% and 13% of the market. What is the probability that the next new book from this publisher captures at most 7.65% of the market?.
6. A jury panel consists of 20 men and 25 women.
a. In how many ways can a jury of 6 men and 6 women be selected
from the panel?
b. If the jury is randomly selected from the panel, what is the
probability that it will have an equal number of men and women?
7. Two people arrive at a train station independently of each other at random times between 1:00 p.m. and 1:30 p.m. What is the probability that one will arrive between 1:00 and 1:10 and the other between 1:15 and 1:30?
8. In statistical surveys where people are selected randomly and are asked questions, experience has shown that only 48% of those under 25 years of age, 67% between 25 and 50, and 89% older than 50 will respond. A social scientist is about to send a questionnaire to a group of randomly selected people. If 25% of the population are younger than 25 and 20% are older than 50, what percent of the group she asks will answer her questionnaire?
9. Ten cards are randomly drawn from an ordinary deck of 52 cards. If exactly four of them are hearts, what is the probability that at least one of the ten cards is a spade?
10. Suppose that 5% of the men and 2% of the women working for a certain corporation make over $130,000 a year. If 40% of the employees of the corporation are women, what percent of those employees who make over $130,000 a year are women?
11. In transmitting Morse code, a certain communication system changes 25% of the dots to dashes and 30% of the dashes to dots. If 40% of the signals sent are dots and 60% are dashes, what is the probability that a dot received was actually sent as a dot?
12. An electronic system fails if both its components fail. Let X be the time (in hours) until the system fails. Experience has shown that
P(X > t) = (1 + t/200)e-t/200 , t * 0.
What is the probability that the system lasts at least 200 but not more than 300 hours?
13. A random variable X has probability density function f(x)
= x/5, x = 1, 2, 3, 4, 5. Find the cumulative distribution function
F(x) of X, and sketch its graph.
14. The annual rainfall (in centimeters) in a certain area is a random
variable with cumulative distribution function
0 x<5
F(x) =
1 - 5/x2 x*5
a. What is the probability that next year it rains at least
6 centimeters? At most 9 centimeters? At least 2 and at most 7 centimeters?
b. What is the probability density function of X?
15. Determine c so that f(x) = cx 2 is a probability density function.
16. A doctor has five patients with migraine headaches. She prescribes for all five a drug that relieves the headaches of 82% of migraine patients. What is the probability that the medicine will not relieve the headaches of exactly two of these patients?
17. Suppose that 10 trains arrive independently at a station every day, each at a random time between 10:00 a.m. and 11:00 a.m. What is the expected number and the variance of those that arrive between 10:15 a.m . and 10:28 a.m.?
18. A certain bank returns bad checks at a Poisson rate of three per day. What is the probability that the bank returns at most four bad checks during the next two days?
19. The policy of the quality control division of a certain corporation is to reject a shipment if more than 5% of its items are defective. A shipment of 500 items is received, 30 of them are randomly selected and tested, and two have been found defective. Should this shipment be rejected?
20. A number is chosen randomly from the interval (0,1). Let X be the resulting random variable. Find the cdf and the pdf of 1/X.
21. a. Check that
2/x3 if x > 1
f(x) =
0 otherwise
is a probability density function.
b. If the random variable X has f(x) (defined in part (a)) as
its pdf, find E(X). What can you say about Var(X)?
Review Problems (Second installment)
22. A random variable X is called exponential if it has pdf of
the form f(x) = ce-cx. If this is the case, then the cdf of X is
F(x) = 1 - e-cx. (In the book, in Example 4.2.4 (p. 175) c is one-over-lambda)
a. Find the mean and variance of X.
b. Find the median of X. (Recall that the median is the number
m with P(X?m) = P(X*m).
c. Find the cdf and pdf of Y = eX.
d. In Question 6.4.4, you proved that the geometric random
variable is memoryless. The appropriate formulation of memoryless
for a continuous random variable is: P(X > s + t | X > t) = P(X > s). (Why?)
Prove that an exponential random variable is memoryless.
23. The lifetime of a tire selected randomly from a used tire shop is 10,000X miles, where X is a random variable with the pdf
2/x2 if 1 < x < 2
f(x) =
0 elsewhere.
a. What percentage of the tires at this shop last fewer than 15,000
miles?
b. What percentage of those having lifetimes less than 15,000
miles last between 10,000 and 12,500 miles?
24. A small college has 1095 students. What is the approximate probability that more than five students from the college were born on New Year’s Day? (You may assume that the birth rates are constant throughout the year and that each year has 365 days.)
25. The ages of subscribers to a certain newspaper are normally distributed
with mean 35.5 years and standard deviation 4.8.
a. What is the probability that the age of a random subscriber
is (i) more than 40 years; (ii) between 30 and 40 years?
b. If you take a random sample of 16 subscribers, what is the
probability that the average age of the people in the sample is (i)
more than 40 years; (ii) between 30 and 40 years?
26. The test scores on a certain standardized exam are normally distributed with mean 500 and standard deviation 100. Find the interval of test scores that is centered at the mean and contains 50% of the scores obtained on the exam.
27. The scores on a certain manual dexterity test are normal with mean 12 and standard deviation 3. If eight randomly selected individuals take the test, what is the probability that none will make a score less than 14?
28. Three cards are drawn at random from an ordinary deck of 2 cards.
If X and Y are the numbers of diamonds and clubs, respectively, calculate
the joint pdf of X and Y.
29. The joint pdf of X and Y is
cxy, 0 ? x ? 1, 0 ? y ? 1
f(x,y) =
0 elsewhere
a. Find c
b. Find the marginal pdf’s of X and Y
c. Are X and Y independent?
d. Find Cov(X,Y)
e. Find E(X + Y)
f. Set up the integrals you would use to do parts (a) - (b) if
the definition of f were altered to be
cxy, 0 ? x ? 1, x + y ? 1
f(x,y) =
0 elsewhere
30. For each of the following problem situations, first decide whether the random variable you are asked to say something about is binomial or a sum of independent, identically distributed random variables. Then use either the DeMoivre-Laplace approximation to the binomial or the Central Limit Theorem to find an approximate answer to the question asked.
a. One hundred bolts are packed in a box. The weight of a bolt has mean 1 oz. and standard deviation 0.1 oz. Find the approximate probability that a box weighs more than 102 oz.
b. A system of components functions if at least 90% of the components function properly. Each component has probability .85 of functioning properly. Find the probability that the system operates.
c. An elevator can carry a maximum of 1650 pounds. The weights of American men aged 18 - 24 are normally distributed with mean 162 lbs and standard deviation 29.1 lbs. What is the probability that 10 American men in this age range will overload the elevator?
d. A psychologist wants to estimate µ, the mean IQ of the students of a certain university. To do so, he takes a sample of size n of the students and measures their IQs. Then he finds the average of these numbers. If he believes that the IQs of these students are independent random variables with variance 170, how big should he choose the sample in order to be 98% sure that his average is accurate within plus or minus 0.2?
e. The length of life of a certain fluorescent fixture has an exponential distribution (see Problem 22) with expected life 10,000 hours. Seventy of these bulbs operate in a factory. Find the probability that at most 40 of them last at least 8000 hours.
f. Light fixtures in a warehouse contain bulbs whose life lengths are exponential (See Problem 22) with a mean of 720 hours. When a light burns out, it is immediately replaced with a new bulb. (i) What is the probability that 3 bulbs last at least 200 hours total? (ii) If we want the probability that the bulbs on hand will last at least 3500 hours to be 0.95, how many bulbs should be stocked?
31. X is a discrete random variable with pdf f(x) = 1/5, x = 1, 2, ..., 5 and zero elsewhere. Find the moment generating function of X.
32. The average number of accidents in a certain intersection
is two per day.
a. What can you say about the probability that at least five
accidents occur tomorrow?
b. If in addition you know that the variance of the number of
accidents per day at the intersection is 2, what can you say about the
probability that at least five accidents occur tomorrow?
33. Describe the Central Limit Theorem
a. intuitively
b. precisely
34. Compare and contrast the Central Limit Theorem and the DeMoivre-Laplace
Theorem.
Hints, Answers, and Solutions to Selected Review Problems
3. If you get confused, you might try a Venn diagram or a chart.
4. a. Remember that the leftmost digit cannot be 0.
b. Try counting numbers by the place furthest to the right where
they have a five.
5. What type of distribution is this talking about?
7. What type of distribution is involved here?
9. Break this up into subproblems.
11. You want (P dot sent | dot received).
17. For each of the ten trains, what is the probability that it will arrive between 10:15 a.m . and 10:28 a.m.?
18. Remember that Poisson rate is not the same as the parameter of a Poisson variable.
20. Start with the definition of the cdf.
22. a. 1/c and 1/c2 (Use integration by parts.)
c. Start out by using the definition of cdf.
24. We have 1095 independent trials (one for each student) , each with
probability of success 1/365 (success = the student was born on New Year’s
Day). Let X be the number of successes. Then X is binomial with parameters
n = 1095 and p = 1/365. Since np and n(1-p) are both *10, it is reasonable
to use the normal approximation to the binomial. The mean of X is np =
3 and the standard deviation is [np(1-p)]1/2 ª 1.73. Thus the probability
that more than five students were born on New Year’s Day is
P(X > 5) ª P((X - 3)/1.73 > (5 - 3)/1.73) ª P(Z >
2/1.73)
= 1 - P(Z ? 1.156)
From tables of normal probabilities, this is approximately 1 - .8760
= .1240
25. Let the random variable X denote the age of a subscriber.
a. (i) We want P(X > 40), which is equal to P((X - 35.5)/4.8 > (40 - 35.5)/4.8). But Z = (X - 35.5)/4.8 is standard normal and (40 - 35.5)/4.8 = .9375, so we want P(Z > .9375) = 1 - P(Z < .9375) ª 1 - .8264 = .1736
b. (i) The age of each of the people in the sample is a random
variable Xi which has the same distribution as X. The random variable Y
in question here is the average of the Xi’s. We want P(Y > 40), which equals
P((Y - 35.5)/ (4.8/4) > (40 - 35.5)/ (4.8/4)).
(Note that since Y has mean 35.5 and standard deviation 4.8/4, what
we have done is just the usual “standardization” process. Why are we dividing
4.8 by 4 to get the standard deviation of Y?). Calculating (40 - 35.5)/
(4.8/4)) = 3.75 and using the Central Limit Theorem, we see that
this is approximately
P( Z > 3.75) = 1 - P( Z > 3.75) ª
1 - 1 = 0.
(Note: 1) “approximately” could be replaced by “equal”, since the random
variables we are averaging are normal, and the sum of normal random variables
is again normal.
2) This illustrates how averaging can change probabilities noticeably.)
29. a. 4 c. Use your answers to part (b)
d. Use your answer to part (c) e. Use Theorem 5.3.3
30. b. 0.21 (your answer may vary slightly depending on just which form
of approximation you use)
d. 23,073. The psychologist has set an unrealistic goal.
e. Y = number of bulbs out of 70 that last at least 8000 hours
is binomial. First calculate P(X * 8000) where X is the life of one bulb.
(See problem 22. You should get 0.449.) Your final answer should be around
0.98.
f. Sum of independent, identically distributed (exponential)
random variables.
32. Use appropriate forms of Chebyshev’s Inequality.