Exercise 9 (#2.22). Let (Yi, Zi), i = 1,...,n, be independent and iden- tically distributed random 2-vectors. The sample correlation coefficient is defined to be T = (n - 1)SYSz SM - D)(Z - Z). where Y = n-15 ZimY, Zen 'Cl_ Z, 8=(m-1)-121,(Y-V), and S2= (n-1)-15 Ziel (Zi- Z) 2. (i) Assume that ELY,| 0 for all r E R and, for any e E O, fo(a) is continuous in r. Let X1 and X2 be independent and identically distributed as fo. Show that if X1 + X, is sufficient for A, then P is an exponential family indexed by 9.Exercise 39 (#2.80). Let X1,..., Xn be random variables with a finite common mean / = EX, and finite variances. Consider the estimation of / under the squared error loss. (i) Show that there is no optimal rule in 3 if 3 contains all possible esti- mators. (ii) Find an optimal rule in n 32 = EX:GER,=1 1=1 i=1 if Var(X,) = 03/a; with an unknown o2 and known a;, i = 1,..., n. (iii) Find an optimal rule in 32 if X1..... X, are identically distributed but 76 Chapter 2. Fundamentals of Statistics are correlated with correlation coefficient p.The class registrations of 120 students are analyzed. It is found that: 30 of the students do not take any of Applied Mechanics, Chemistry, or Computers. 15 of them take only Applied Mechanics. 25 of them take Chemistry and Computers but not Applied Mechanics. 20 of them take Applied Mechanics and Computers but not Chemistry. 14 Basic Probability 10 of them take all three of Applied Mechanics, Chemistry, and Computers. A total of 45 of them take Chemistry. 5 of them take only Chemistry. a) How many of the students take Applied Mechanics and Chemistry but not Computers? b) How many of the students take only Computers? c) What is the total number of students taking Computers? d) If a student is chosen at random from those who take neither Chemistry nor Computers, what is the probability that he or she does not take Applied Mechanics either? e) If one of the students who take at least two of the three courses is chosen at random, what is the probability that he or she takes all three courses?A bag contains 6 red ballsI 5 yellow balls and 3 green balls. A ball is drawn at random. What is the probability that the ball is: {a} green, {b} not yellow, {c} red or yellow? A pilot plant has produced metallurgical batches which are summarized as follows : Luz-w strength High strength Low in impurities 2 27 High in impurities 12 4 If them results are representative of fullscale production, nd estimated probabilities that a production batch will be: i} low in impurities ii) high strength iii} both high in impurities and high strength iv} both high in impurities and low strength 1\" Basic Probability 3. If the numbers of dots on the upward faces of two standard sixsided dice give the score for that throw, what is the probability of making a score of 7 in one throw of a pair of fair dice? 4. In each of the following cases detennine a decimal value for the probability of the event: a} the fair odds against a successful oil well are ltol. b} the fair odds that a bid will succeed are lto. 5. Two nuts having LLB. coarse threads and three nuts having U3. ne threads are mixed accidentally with four nuts having metric threads. The nuts are otherwise identical. A nut is chosen at random. a} What is the probability it has U3. coarse threads? b} What is the probability that its threads are not metric? c} If the rst nut has US. coarse threads, what is the probability that a second nut chosen at random has metric threads? d} If you are repairing a car engine and accidentally replace one type of nut with another when you put the engine back together, very briey, what may be the consequences? {a} How many different positive threedigit whole numbers can be formed from the four digits 2, t5, 7, and 9 ifany digit can be repeated? {b} How many different positive whole numbers less than limit can be formed from 2, 6-. T, 9 ifany digit can be repeated? {c} How many numbers in part (b) are less than {Sgt} (Le. up to 6'39}? {d} What is the probability that a positive whole number less than I'll}, chosen at random from 2. 6. 7. 9 and allowing any digit to be repeatedI will be less than 63a? Answer question 7 again for the case where the digits 2, 6, 7, '9 can not be repeated. For each of the following. determine {i} the probability of each eventI {ii} the fair odds against each eventI and {iii} the fair odds in favour of each event: {a} a ve appears in the toss of a fair sixsided die. {b} a red jack appears in draw of a single card from a wellshuffled Sillcard bridge deck
