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Q1:13. Suppose that the joint probability density of two random variables X1 and X2 is given by: f(X1,X2)= (6e-2X1-3x2 for x1 > 0 and x2 > 0, 0 otherwise Find the following probabilities. Show all steps. a) P(1≤X1≤2,2≤X2≤3) b) P(X1<2, X2>2). c) Find the marginal density fi(x1).See Answer
Q2:12. Let X and Y have joint pdf: f(x,y) =12x(1-x)y, 0<x<1, and 0<y< 1 a) Find the marginal densities of X and Y. b) Determine the conditional pdf f1 (x|y). c) Find E(X) and E(Y). d) Find P(X ≤ 0.5|Y = 0.5). e) Determine if X and Y are independent.See Answer
Q3:6. Suppose that the annual amount of rainfall (in million tons) accumulated in a lake follows a gamma distribution with a = 3 and ß = 5. a) Find the expected annual rainfall accumulated in this lake. b) Find the standard deviation of the annual amount of rainfall in this lake. c) Find P(X < 4), use R. d) Find P(2<X <5, use R.See Answer
Q4:Problem 4. The following data are the temperatures (in degree Fahrenheit) of effluent at discharge from a sewage treatment facility on consecutive days: 43 47 51 48 52 50 46 49 45 52 46 51 44 49 46 51 59 45 44 50 48 50 49 50 Using the Method of Moments, fit normal, lognormal distribution, 2-parameter exponential, and uniform distributions to the data and provide the following for each distribution: a. Parameters of the distribution. b. Using the fitted distribution, find the probability that temperature exceeds 58 °F? 1See Answer
Q5:Problem 1. Exercise 4.10.8 in Applied Statistics & Probability for Engineers, 7th Ed. An article in Journal of Hydrology titled "Use of a Lognormal Distribution Model for Estimating Soil Water Retention Curves from Particle-Size Distribution Data" (2006, Vol. 323(1), pp. 325-334) considered a lognormal distribution model to estimate water retention curves for a range of soil textures. The particle-size distribution (in centimeters) was modeled as a lognormal random variable X with 0 = -3.8 and w = 0.7. Determine the following: a. P(X < 0.02) b. Value for x such that P(X ≤ x) = 0.95 c. Mean and variance of XSee Answer
Q6:Problem 3. A nuclear facility in a coastal region is built to withstand ocean wave forces. Suppose the annual maximum wave height of the ocean waves (above the sea level) is a Gamma distributed random variable with mean height of 4.0 meters and coefficient of variation of 0.8. a. What is the probability that the wave height will exceed 6 meters in any year? b. To construct the nuclear facility in a safe manner, an engineer is assigned to compute a design wave height (above sea level) that will not be exceeded by ocean waves over a 10- year period with a probability of 95%, i.e., 10-yr reliability = 0.95. Assuming that wave height exceedances between years are statistically independent, what should be the height of the design wave above the sea level?See Answer
Q7:The distance between major cracks in a highway follows a [1-parameter] exponential distribution with a mean of 5 miles. a. What is the probability that there are no major cracks in a 10-mile stretch of the highway? b. What is the standard deviation of the distance between major cracks? c. What is the probability that the first major crack occurs between 12 and 15 miles of the start of inspection? d. Given that there are no cracks in the first 5 miles inspected, what is the probability that there are no major cracks in the next 10 miles inspected?See Answer
Q8:7. A printer company claims that the mean warm-up time of a certain brand of printer is at most 10 seconds. An engineer of another company is conducting a statistical test to show this is an underestimate, and the mean warm-up time is more than 10 seconds. a) State the testing hypothesis. b) The test yielded a p-value of 0.035. What would be the decision of the test if α = 0.05? c) Suppose a further study establishes that the true mean warm-up time is 9 seconds. Did the engineer make the correct decision in part (b)? If not, what type of error did he/she make? 8. The incubation period of respiratory infection is known to have a normal distribution with a mean of 8 days and a standard deviation of 3 days. Suppose a group of researchers claimed that the true mean incubation period is shorter than 8 days. A test is conducted using a random sample of 20 patients. a) Formulate hypotheses for this test. b) Consider a rejection region (X ≤6}. Suppose the test failed to reject Ho. If the true mean incubation period is 5 days, what is the Type II error probability of the test using σ =3?See Answer
Q9:Problem 4 Playing Cards Suppose you have a deok of 52 playing cards and you draw 2 cards at random (without replacement). a. The probability that you will draw two cards of the same rank (e.g., two 7s, two Kings, etc.) from a standard deck of 52 cards is 13x¹C₂ ≈ 0.0588 Perform a data generation process and verify that the simulated probability is nearly close to this value for a sufficiently large simulation runs. Explain each code line and its output. b. Suppose you draw 5 cards at random (without replacement). What is the probability that you will draw a full house OR four of a kind? Reminder: A full house means 3 of one value card and 2 of another value card (for example, three kings and two 8s); four of a kind means 4 cards of the same value. NOTE: even if you can't fully solve this, describe the approach you would take and what principles or theorems you might make use of, and make as much progress as you can.See Answer
Q10:Question 1 of 5 Problem 1 Selling Tickets and Making Change A university event requires a fifty dirham "donation" per attendee. Assume that every person who comes to the event pays with cash and each one has either a 50 Dhs bill or a 100 Dhs bill. The team that collects payment at the door neglected to get any small bills to be able to make change. If at least as many attendees pay with 50 as pay with 100 then the ticket sellers will not have to issue anyone an "IOU" (I owe you) and send them their change later. Suppose 2n people come to the event and every individual pays for their own "ticket" and that by the end of the evening there were exactly n with 50 Dhs notes and exactly n with 100 Dhs notes. We want to think about different the implications of them arriving in different orders. For example, if all the people with 100s arrived first then we would need to issue 50 IOUS. 1. What problem that you have seen in your pre-class work does this problem bear a resemblance to? 2. How can you use that analogous problem to give a geometric interpretation of sequences of arrivals of people with 50s and people with 100s that never require an IOU to be issued.See Answer
Q11:Problem 2. Exercise 4.1.9 in Applied Statistics & Probability for Engineers, 7th Ed. The waiting time for service at a hospital emergency department (in hours) follows a distribution with probability density function f(x) = 0.5 exp(-0.5x) for 0 < x. Determine the following: a. P(X<0.5) b. P(X>2)See Answer
Q12:Pr 4. [30 pts] Exercise 2.8.3 in Applied Statistics & Probability for Engineers, 7th Ed. A new analytical method to detect pollutants in water is being tested. This new method of chemical analysis is important because, if adopted, it could be used to detect three different contaminants-organic pollutants, volatile solvents, and chlorinated compounds-instead of having to use a single test for each pollutant. The makers of the test claim that it can detect high levels of organic pollutants with 99.7% accuracy, volatile solvents with 99.95% accuracy, and chlorinated compounds with 89.7% accuracy. If a pollutant is not present, the test does not signal. Samples are prepared for the calibration of the test and 60% of them are contaminated with organic pollutants, 27% with volatile solvents, and 13% with traces of chlorinated compounds. A test sample is selected randomly. a) What is the probability that the test will signal? b) If the test signals, what is the probability that chlorinated compounds are present?See Answer
Q13:Pr 3. [10 pts] The probability of failure of a wastewater treatment facility is normally 0.02. During significant power outages, the probability of failure of the system is 0.28. The probability of significant power outage in the area is 0.04. What is the total probability of failure of the facility?See Answer
Q14:Pr 2. [25 pts] Exercise 2.3.11 in Applied Statistics & Probability for Engineers, 7th Ed.: Samples of emissions from three suppliers are classified for conformance to air-quality specifications. The results from 100 samples are summarized as follows: Supplier 1 2 3 Conform No 8 5 10 e) P(AUB) f) P(A'n B) Yes 22 25 30 Let A denote the event that a sample is from supplier 1, and let B denote the event that a sample conforms to specifications. If a sample is selected at random, determine the following probabilities: a) P(A) b) P(B) c) P(A¹) d) P(An B)See Answer
Q15:RISK 4. MAXIMIZE RETURN/MINIMIZE You are considering the risk-return profile of two mutual funds for investment. The relatively risky fund promises an expected return of 8% with a standard deviation of 14%. The relatively less risky fund promises an expected return of 4% with a standard deviation of 5%. Assume that the returns are approximately normally distributes. a) Which mutual fund you will pick if your objective is to minimize the probability of earning a negative return? b) Which mutual fund will you pick if your objective is to maximize the probability of earning a return above 8%?See Answer
Q16:3. MPG RATING OF CARS Suppose that the miles-per-gallon rating of passenger cars is a normally distributed random variable with a mean of 33.8 mpg and standard deviation of 3.5 mpg. a) What is the probability that a randomly selected passenger car gets at least 40 mpg. b) What is the probability that a randomly selected passenger car gets at most 30 mpg. c) What is the probability that a randomly selected passenger car gets between 30 and 40 mpg. d) An automobile manufacturer wants to build a new passenger car with an mpg rating that improves upon 99% of existing cars. What is the minimum mpg that would achieve this goal? DICKSee Answer
Q17:2. CHEATING ON AN EXAM A professor has learned that three students in her class of 20 will cheat on the exam. She decides to focus her attention on four randomly chosen students during the exam. a) Find the probability that she finds at least one of the students cheating if she focuses on four randomly chosen students? b) Find the probability that she finds at least one of the students cheating if she focuses on six randomly chosen students?See Answer
Q18:1. UNDERWATER MORTGAGE Twenty percent of US mortgages are "underwater" (The Boston Globe, March 5, 2009). A mortgage is considered underwater if the value of the home is less than what is owed on the mortgage. Suppose 100 mortgage holders are randomly selected. a) What is the probability that more than 20 of the mortgages are underwater? b) What is the probability that at least 25 of the mortgages are underwater?See Answer
Q19:Question 4 Let the random variable Z follow a standard normal distribution. Find the following probabilities. a) (4 points) P(Z<1.72), b) (4 points) P(Z>1.1), c) (4 points) P(1.1<Z<1.72), d) (4 points) P(Z<-1.1), e) (4 points) P(-0.23<Z$2.00).See Answer
Q20:Question 2 In a typical month, an insurance agent presents life insurance plans to 40 potential customers. Historically, one in four such customers chooses to buy life insurance from this agent. Based on the relevant Binomial distribution, answer the following questions about X, the number of customers who will buy life insurance from this agent in the coming month: a) (5 points) What is the probability that X is exactly 5? 2 OPRE 3360- Managerial Methods in Decision Making Under Uncertainty b) (5 points) What is the probability that X is no more than 10? c) (5 points) What is the probability that X is at least 20? d) (5 points) Determine the mean and standard deviation of X. Homework 2See Answer

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