statistics
Leila@
re_.zipHW4_STAT205.pdf
Statistical Inference I: J. Lee Assignment 4
Problem 1. Ten years ago at a certain insurance company, the size of claims under homeowner insurance policies had an exponential distribution. Furthermore, 25% of claims were less than $1000. Today, the size of claims still has an exponential distribution but, owing to inflation, every claim made today is twice the size of a similar claim made 10 years ago. Determine the probability that a claim made today is less than $1000.
Problem 2. The lifetime (in hours) of a lightbulb is an exponentially distributed random variable with parameter λ = 0.01 (units of hours−1).
(a) What is the probability that the light bulb is still burning one week after it is installed?
(b) Assume that the bulb was installed at noon today and assume that at 3:00pm tomorrow you notice that the bulb is still working. What is the chance that the bulb will burn out at some time between 4:30pm and 6:00pm tomorrow?
Problem 3. Suppose that X has density fX (x) given below.
fX (x) =
{ x/2 if 0 < x < 2 0 otherwise.
Let Y = 1
4X + 2 .
(a) Compute the cdf and the density of Y . (Remember to be very explicit about all cases!) Plot them.
(b) Compute the variance of Y and of Y 2.
Problem 4. Suppose that X is a normal random variable with parameters µ = 5,σ2 = 49. Using the table of the normal distribution (Z-table, in the folder of Week 4 in BB), compute:
(a) P(X > 5.5)
(b) P(4 < X < 6.5)
(c) P(X < 8)
(d) P(|X − 7| ≥ 4)
Problem 5. When you put your money into a soft drink machine at the Union, a paper cup comes down, and the drink is dispensed into the cup. You are supposed to get 8 oz of drink. However, the actual amount of drink dispensed is random, having a normal distribution with µ equal to the machine setting and σ = 0.25 oz. What should the machine setting be so that, in the long run, only 2% of the drinks will contain less that 8 oz?
Problem 6. The time between the first and second heart attacks for a certain group of people is an exponential random variable. If 50% of those who have had a heart attack will have another one within the next five years, what is the probability that a person who had one heart attack five years ago will not have another one in the next five years?
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LP Additional Practice for Exam.pdf
OPR 320: Additional Practice Questions for the Exam Key Elements in LP Sensitivity Report by Excel
1. Shadow price of a constraint is the change in the objective for a unit increase of the R.H.S. of the constraint (e.g., the amount of resource available).
2. The shadow price and the binding constraints remain the same as long as the R.H.S. changes within allowable increase and allowable decrease.
3. The optimal solution remains the same as long as the objective coefficients change within allowable increase and allowable decrease.
4. Always keep units consistent! Practice 1 You are managing a small consulting company. You have two consultants working for you (A and B), and three client companies (1, 2, and 3). You have negotiated a different per-hour payment rate with each of the three companies ($100 for client 1, $150 for client 2, and $200 for client 3). Each company has imposed an upper limit of 60 billable hours per week. The two consultants are willing to work 70-hour weeks, and can work on either of the projects. Consultant B is more experienced and is paid $90 per hour, while consultant A is paid $80 per hour (both independent of the project that they are assigned to). Your objective is to maximize the profit.
a. Formulate the problem as an LP.
b. If Consultant A falls sick and is able to work for only 60 hours this week, how much do you lose in profits?
c. If Consultant B agrees to work for another 10 hours this week, what is the maximum you would be willing to pay her per hour (for each additional hour worked)? Justify your answer.
d. What if Consultant A only works for 50 hours? With the given information, could you tell exactly how much you lose in profits? Justify your answers.
e. What is the shadow price of Client 1’s billable hours constraint (blank box in table)? Why? Practice 2 A leading electronics firm manufactures and sells 3 types of high-quality headphones at its facility in Malaysia. Variable costs, revenues and weekly demands are given below. Weekly demands are the upper limits on what can be sold. Also provided are the production times per headphone and the capacity for each manufacturing process (in minutes). The firm’s objective is to maximize profit.
a. Formulate the problem as an LP:
Type Price Cost Demand Fab Assbly Pckg
DJ 80 47 45 20 35 9 Pro 70 40 150 10 28 7
Elite 50 27 62 17 15 5 Capacity 2000 5500 910
The output for an LP solution is provided below.
b. By how much would total contribution increase if Assembly capacity is increased by 1 hour?
c. By how much would total contribution increase if Packaging capacity is increased by 1 hour?
d. If demand for only one type of headphone could be increased by a new marketing campaign, which type of headphone should be advertised? What is the maximum amount that should be spent to generate one additional unit of demand for that type?
e. In addition to manufacturing in-house, the firm subcontracts the Packaging of ‘Professional’ headphones to a small manufacturer – Earblast, Inc. What is the maximum that Earblast, Inc. should be paid per Professional headphone?
Practice_Midterm.pdf
Practice Midterm Statistical Inference I, Dr. Jinwook Lee
Problem 1. I tell you that, for two events A and B, P(A) = 4/5,P(B|A) = 1/2, and that A and B are independent. For each statement, say “True”, “False”, or “Can’t Tell”, and give a reason for your answer.
(a) A and B are mutually exclusive
(b) A and A∪B are independent
(c) P(B) = P(A|B)
(d) P(A|B) < P(B|A)
(e) P(B) ≤ P(A)
Problem 2.
(a) A gambler has in his pocket “two” fair coins and “two” two-headed coin. He selects one of the coins at random; when he flips it, it shows heads. What is the probability that it is a fair coin?
(b) Suppose that he flips the same coin a second time and again it shows heads. What is now the probability that it is a fair coin?
(c) Suppose that he flips the same coin a third time and it shows tails. What is now the probability that it is a fair coin?
Problem 3. Consider a random variable X whose probability mass function (pmf) is given by
p(x) =
0.1 if x = −3 0.2 if x = 0 0.3 if x = 2.2 p if x = 3 3p if x = 4 0 otherwise.
(a) What is p?
(b) Compute P(X2 − 2 > 6).
(c) What is F(0)? What is F(1)? What is F(F(3.1))? (Here, F(·) denotes the distribution function (cdf) for X)
(d) Sketch a plot of the function F(x). (Make sure to label the coordinates on the axes!)
(e) What is P(2X − 3 ≤ 2 | X ≥ 2.1)?
(f) Compute E(X).
(g) Compute E(F(X)).
Problem 4. Let X denote the length (in meters) of one side of a square sheet of plywood. Assume that the density function of X is given by
f(x) =
x if 0 ≤ x < 1 1/2 if 1 ≤ x < 2 0 otherwise.
(a) On average, what fraction of squares of plywood have side length longer than 75cm?
(b) Let A denote the area (in square meters) of the sheet of plywood. Compute Var(A).
(c) Let Y = 5X. Compute the cdf, FY (y), of Y . Be very explicit! You must show the value of FY (y) for all values of y; be careful about all cases. (You may find it helpful to sketch a plot of FY (y), but it is not required.)
(d) Find the pdf, fY (y), of Y . Be very explicit! You must show the value of fY (y) for all values of y; be careful about all cases. (You may find it helpful to sketch a plot of fY (y), but it is not required.)
Problem 5. Let X denote the lifetime of a radio, in years, manufactured by a certain company. Assume that the lifetime is exponentially distributed, with a mean of 5 years.
(a) What is the probability p that a radio lasts more than 15 years?
(b) What is the (exact) probability that, of 1000 such radios that you purchase, at least four of them last more than 15 years? (You can leave your expression with a “p” in it, where p is the answer from part (a); you do not need to evaluate it.)
(c) Give an alternative expression that approximates your answer to part (b). Justify your answer: What distribution are you using to obtain the approximation?
Problem 6. The length of an aluminum-coated steel sheet manufactured by a certain factory is approxi- mately normal with mean 75 centimeters and standard deviation 2 centimeters. Find the probability that a randomly selected sheet manufactured by this factory is between 74.2 and 75.6 centimeters.
Problem 7. People enter a store, on average one every two minutes.
(a) What is the probability that no people enter between 12:00 and 12:05?
(b) Find the probability that at least 4 people enter during [12:00,12:05].
Problem 8. The suicide rate in a certain state is 1 suicide per 100,000 inhabitants per month.
(a) Find the probability that in a city of 400,000 inhabitants within this state, there will be 8 or more suicides in the month of June this year.
(b) What is the probability that there will be at least two months during the year in which the city has 8 or more suicides?
(c) Counting the present month as month number 1, what is the probability that the first month to have 8 or more suicides (in the city) will be month number i, i ≥ 1?
(d) Today is January 1. What is the probability that the fourth month to have 8 or more suicides in the city is December?
Problem 9. The donuts you make in your bakery have random sizes. The radius (in inches) of a (circular) donut is a random variable whose density function is given by
f(x) =
x2 if 0 ≤ x < 1 1 if 1 ≤ x < 5/3 0 otherwise.
(a) What fraction of donuts have radius larger than 1/2 inch?
(b) What is the expected radius, µ, of a donut? What is the probability that a donut has radius equal to µ?
(c) Compute the variance of the radius, X.
(d) Let Y denote the diameter of a donut. Compute the cdf, FY (y), of Y. Be very explicit! You must show the value of FY (y) for all values of y; be careful about all cases.
(e) Find the pdf, fY (y), of Y . Be very explicit! You must show the value of fY (y) for all values of y; be careful about all cases. (You may find it helpful to sketch a plot of fY (y), but it is not required.)
