Essay About Probabilities Of Another Event And Random Variable X

Statistic Exam Question Answers
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(5%) Given: The probabilities of three events, A, B, and C, occurring areP(A) = 0.35, P(B) = 0.45, and P(C) = 0.2. Assuming that A, B, or C hasoccurred, the probabilities of another event, X, occurring are P(X│A)=0.8,P(X│B)=0.65, and P(X│C)=0.3. Find P(A│X)= _____________.2. (5%) A Ph.D. graduate has applied for a job with two universities: A and B.The graduate feels that she has a 60% chance of receiving an offer fromuniversity A and a 50% chance of receiving an offer from university B. If shereceives an offer from university B, she believes that she has an 80% chanceof receiving an offer from university A. What is the probability that atleast one university will make her an offer?Probability = _________________3. (5%) The monthly sales at a bookstore have a mean of 50,000 and a standarddeviation of 6,000. Profits are calculated by multiplying sales by 40% andsubtracting fixed costs of 12,000. Find the standard deviation of monthly profits.Standard deviation of monthly profits = ____________4. (5%) Suppose a disease is present in 3% of population. A diagnostic test forsuch disease shows 10% false positive and 5% false negative. That is, for apatient having the disease,, the test shows positive (+) with probability0.95 and negative (-) with probability 0.05. For a patient not havingdisease, the test shows positive (+) with probability 0.10 and negative (-)with probability 0.90. If a patient’s test show (+), what’s the probabilityof his having the disease?Probability = ______________5. (6%) Throw a dice n times fairly and observe the number of dice each time.Assume a random variable X is the frequency that the number of dice is one.(3%) Write down the probability distribution of random variableb. (3%) Assume n = 5, P(X<=1 or X>=5)=______________6. (5%) A communication system consists of n components, each of which will,independently, function with probability p. The total system will be able tooperate effectively if at least one-half of its components function. For whatvalues of p is a 5-component system more likely to operate effectively than a3-component system?Values of p should be ___________7. (6%) Suppose that earthquake occur in the eastern part of Taiwan inaccordance with the assumptions for the Poisson probability distribution at a

rate of 2 per day.a. (3%) Find the probability that at least 3 earthquake occur during the next   2 days = _______________b. (3%) Find the probability distribution of the time, starting from now,   until the next earthquake. _____________________8. (5%) Suppose that the proportion of colorblind people in a certainpopulation is 0.005. What is the probability that there will not be more thanone colorblind person in a randomly chosen group of 600 people? You can usePoisson distribution to approximate the binomial distribution.Probability = __________________9. (5%) Suppose the daily amount of waste generated per person is normallydistributed, with mean 3.58 pounds and standard deviation 1.04 pounds. Of thedaily amounts of waste generated per person, 67.72% would be greater thanwhat amount? ____________________10. (5%) Suppose that customers arrive at a drive through window at anaverage rate of three customers per minute and that their arrival follow thePoisson model. Use the appropriate distribution to find the probability thatthe next customer will arrive within 1.5 minutes.Probability = ________________11. (7%) Suppose that patrons of a restaurant were asked whether they preferredbeer or whether they preferred wine. 60% said that they preferred beer. 70% ofthe patrons were male. 80% of the males preferred beer.(3%) What is the probability a randomly selected patron prefers wine?Probability = ____________b. (3%) Suppose a randomly selected patron is a female. What is the probabilitythat the patron prefers wine?Probability = ____________c. (1%) Are gender of patrons and drinking preference independent?Choose yes or no = _____________  Explain below.12. (5%) Suppose that X is a random variable for which E(X)=μ and Var(X)=σ^2.Find E[X(X-1)]=__________________13. (5%) Suppose that X and Y are random variables such that Var(X)=9, Var(Y)=4,and ρ(X,Y)=-1/6. Find Var(X-3Y+4)=______________   14. (5%) The average score of all student in an economics class has a mean of70 and a standard deviation of 3. Suppose a sample of 36 students took theclass this semester. Find the probability that the average score of the 36