Problems for Final Exam Please note that the problems in the exam can be reformulated or slightly altered. CHAPTER IV 1. Let and be two r.v.'s with the joint pdf ? ? elsewhere. ? ?? ? ? ?? ? ?? ? ? ??...

Problems for Final Exam Please note that the problems in the exam can be reformulated or slightly altered. CHAPTER IV 1. Let and be two r.v.'s with the joint pdf ? ? elsewhere. ? ?? ? ? ?? ? ?? ? ? ?? ? ? ? ? ? ????? ?? Calculate ??? ??? ? ?? ? ? ?? ???? ??? ? ? ? 2. Let and be two r.v.'s with the joint pdf ? ? elsewhere. ???? ?? ? ??? ? ??? ? ? ? ? ? ? ? ?? ? ? ? ??? Determine the value of the constant . ? ???? Are and independent? ? ? 3. Let and be two r.v.'s with the joint pdf ? ? elsewhere ???? ?? ? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ? ? ? ??? ?? ? ? ? where is a positive constant. Are the events and independent? ? ?? ? ?? ? ? ? ? ?? ? ? 4. Let be iid (independent, identically distributed) r.v.'s, each Gaussian with ??? ?? ?? parameters and . Find the pdf of the sample mean (see _ ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? Example 2.2) and identify it. : Use the mgf technique. Hint 2 5. Let and be two independent r.v.'s with their respective mgf's ? ? ??? ? ? ? and ? ?? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? 5 5 ? 3 Calculate ??? ???? ? ?? ???? ???? ? ???? ? ???. ? ? ????? Var . ??? ? ?? ? ?? 6. Let and be gamma distributed with parameters and . Find the pdf ? ? ???? ?? ???? ?? of . ? ? ? 7. Suppose two fair dice are rolled and let and denote the outcome of the first and ?? ?? second die, respectively. Prove that the r.v.'s and are ? ? ?? ? ??? ? ? ??? ? ?? uncorrelated. Chapter IV, section 6 and Chapter V, sections XXXXXXXXXXThe number of students enrolled in calculus classes at FIT is a Poisson r.v. with parameter . Use the CLT approximation to find the probability that the new ? ? ??? enrollment is going to be 120 or more students. 9. If is a gamma r.v. with parameters (i.e. -Erlang r.v. with , ? ??? ?? ? ? ? ?? approximately how large need be in order that ? ? ??? ? ?? ? ????? ? ???? ? ? 10. Suppose we need to estimate the proportion of traffic violators in some area. ? Assuming that in a sample of observations there were violations find the maximal ??? ?? confidence interval for with the confidence of 95%. Find the maximal confidence ? interval for with confidence 95% if it is known that . ? ? ? ??? 11. Let , where is an exponential r.v. with parameter unknown. ??? ???? ?? ? ??? ? ? Find the m.l.e. and MLE of . Give an unbiased and consistent estimator of . ? ? ? 3 12. Let the joint pdf of and be given by ? ? elsewhere. ???? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ? ? ? ? ???? ?? Find the conditional pdf and . ?????? ??? ? ??? ? ?? 13. If X? is drawn from an exponential population with parameter and the prior of ? ? ? ? is gamma with parameters , then show that the posterior of is also gamma with ??? ?? ? parameters . Find the Bayes estimator of . ?? ? ?? ? ? ?? ? ? ??? ? ? . 14. Let be the proportion of defective items in a large manufactured lot, which is ? unknown, but its prior distribution is supposed to be the standard uniform. Let a random sample of items, be drawn and suppose just one of them turned ?? X?? ? ???? ?? ???? out to be defective. Find the posterior density and give the ????x??? ? ??????? ?? ???? Bayes estimate of . ? 15. Assume that the prior of the proportion of the defective items is beta with ? parameters and . Suppose a sample of items was drawn and that ? ? ? ? ? ?? ? ? ? ?? were defective. Find the posterior pdf of . Give the Bayes estimator of and find out if ? ? it is consistent. 16. The number of connections to a wrong phone number is modeled by a Poisson distribution. Suppose the prior of its parameter is known to be gamma with parameters ? ? and . Estimate (the mean number of wrong connections) by observing a sample ? ? ??? ?? ?? of wrong connections on different days. Find the Bayes estimator of . ? ? CHAPTER XI. 17. Let . Prove that ? ? ????? ??? ? ? ? ?? ? ? ?? 18. A coin is thrown until a head occurs and the number of tosses is recorded. After ? repeating the experiment 256 times the following results were obtained: ?? ? ? ? ? ? ? ? ? ? ??? ?? ?? ?? ? ? ? ? ? ? 4 where is the number of tosses needed to obtain the first head and is the number of ??? ?? experiments in which occurs. Test the hypothesis at the significance that the ??? ???? observed distribution of is geometric with parameter ? ? ? ? ? ? 19. According to the Mendelian theory of genetics a certain garden pea plant should produce either white, or pink, or red flowers, with respective probabilities To test ? ? ? ? ? ? ? ? ? this theory, a sample of 564 peas were studied and they produced white, 291 pink, ??? and 132 red flowers. Test the hypothesis at significance that the observed sample ???? agrees with Mendelian theory. Also give the P-value. Answer: and P-value ? ? ?????? ? ?????. 20. It is conjectured that the daily number of electrical power failures in a certain city obeys the Poisson law with mean 4.2. A total of 150 days of observations produced the following results: # Failures Days Observed ? ? ? ? ? ?? ? ?? ? ?? ? ?? ? ?? ? ?? ? ? ? ? ?? ? ?? ? Test the H that the number of power outages is indeed Poisson with at ? ? ? ??? ? ? ???? and give the P-value. Answer: P-value . ? ? ??????? ? ????? 21. A random sample of 795 individuals was collected to investigate whether smoking and drinking alcohol are related. The results were as follows: 5 Heavy Smoker Moderate Smoker Nonsmoker Heavy Drinker Moderate Drinker Nondrinker ?? ?? ? ?? ? ??? ?? ?? ??? Test the hypothesis that drinking alcohol and smoking are independent at . ? ? ???? 22. Suppose 300 people were selected at random and each person in the sample is classified according to blood type, They were also classified according to ?? ?? ?? ??? other blood types positive or negative. The observed data are put in the following ?? table: ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ? positive negative Test the hypothesis that the two classifications of blood types are independent at ? ? ???? and also find the P-value.