7. (BH 7.63) There will be X Pois() courses oered at a certain school next year. (a) Find the expected number of choices of 4 courses (in terms of , fully simplied), assuming that simultaneous enrollment is allowed if there are time conicts. (b) Now suppose that simultaneous enrollment is not allowed. Suppose that most faculty only want to teach on Tuesdays and Thursdays, and most students only want to take courses that start at 10 am or later, and as a result there are only four possible time slots: 10 am, 11:30 am, 1 pm, 2:30 pm (each course meets Tuesday-Thursday for an hour and a half, starting at one of these times). Rather than trying to avoid major conicts, the school schedules the courses completely randomly: after the list of courses for next year is determined, they randomly get assigned to time slots, independently and with probability 1/4 for each time slot. Let Xam and Xpm be the number of morning and afternoon courses for next year, respectively (where \"morning\" means starting before noon). Find the joint PMF of Xam and Xpm , i.e., nd P (Xam = a, Xpm = b) for all a, b. (c) Continuing as in (b), let X1 , X2 , X3 , X4 be the number of 10 am, 11:30 am, 1 pm, 2:30 pm courses for next year, respectively. What is the joint distribution of X1 , X2 , X3 , X4 ? (The result is completely analogous to that of Xam , Xpm ; you can derive it by thinking conditionally, but for this part you are also allowed to just use the fact that the result is analogous to that of (b).) Use this to nd the expected number of choices of 4 non-conicting courses (in terms of , fully simplied). What is the ratio of the expected value from (a) to this expected value? 8. (BH 7.69) Let X be the number of statistics majors in a certain college in the Class of 2030, viewed as an r.v. Each statistics major chooses between two tracks: a general track in statistical principles and methods, and a track in quantitative nance. Suppose that each statistics major chooses randomly which of these two tracks to follow, independently, with probability p of choosing the general track. Let Y be the number of statistics majors who choose the general track, and Z be the number of statistics majors who choose the quantitative nance track. (a) Suppose that X Pois(). (This isn't the exact distribution in reality since a Poisson is unbounded, but it may be a very good approximation.) Find the correlation between X and Y . (b) Let n be the size of the Class of 2030, where n is a known constant. For this part and the next, instead of assuming that X is Poisson, assume that each of the n students chooses to be a statistics major with probability r, independently. Find the joint distribution of Y , Z, and the number of non-statistics majors, and their marginal distributions. (c) Continuing as in (b), nd the correlation between X and Y . Answer 8 a) Let X denote the number of stats major Let Y denote the stats major who chose the general track in statistical principles Let Z denote the stats major who chose the quantitative finance track Consider the event of number of stats major as a mutually exclusive and exhaustive events i.e X = Y + Z Since X ~ Poisson () then using the additive property of independent poisson variates (using the mgf or characteristic function property of additive nature ) We can say that Y & Z also follow poisson distribution. i.e If X=Y+Z ~ Poisson() , then Y & Z ~ Poisson(/2) each respectively . So the correlation : corr ( X, Y) = corr ( Y+Z, Y) = corr( Y,Y) + corr(Z,Y) = 1 + corr( Z,Y) Answer 8 a) Let X denote the number of stats major Let Y denote the stats major who chose the general track in statistical principles Let Z denote the stats major who chose the quantitative finance track Consider the event of number of stats major as a mutually exclusive and exhaustive events i.e X = Y + Z Since X ~ Poisson () then using the additive property of independent poisson variates (using the mgf or characteristic function property of additive nature ) We can say that Y & Z also follow poisson distribution. i.e If X=Y+Z ~ Poisson() , then Y & Z ~ Poisson(/2) each respectively . So the correlation : corr ( X, Y) = corr ( Y+Z, Y) = corr( Y,Y) + corr(Z,Y) = 1 + corr( Z,Y) b) In this case we assume that X which denotes the number of stats major , X = n and P(X=n) = r chosen in an independent manner . We know that X = Y+ Z n = Y + Z Y and Z are independent Poisson variates with parameters 2 , 2 . ( from part a) X = n is a binomial variate with parameters (n , r) then the mgf of X (where X=Y+Z ) t t Mn (t) = ( q + p e n = ( 1+ ( e -1)p )n here p = r lim M n(t) r = / n and let n we get n If we take nr = lim M n (t ) n = ( lim 1+ n (et 1) n n ) distribution with parameter . Hence by uniqueness theorem of mgf of X fixed . t = exp ( ( e 1) which is the mgf of Poisson P () as n with nr = is