5. (BH 7.63) There will be X~Pois(A) courses offered at a certain school next ye

Question

5. (BH 7.63) There will be X~Pois(A) courses offered at a certain school next year (a) Find the expected number of choices of 4 courses (in terms of , fully simplified), assuming that simultaneous enrollment is allowed if there are time conflicts. (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 conflicts, 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, respec- tively (where "morning" means starting before noon). Find the joint PMF of Xam and Xpm, i.e., find P(Xam a, Xpm b) for all a, b. (c) Continuing as in (b), let Xi, 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 Xi, X2, X3, X4? (The result is completely analogous to that of Xam, Xpmi 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 find the expected number of choices of 4 non- conflicting courses (in terms of , fully simplified), what is the ratio of the expected value from (a) to this expected value?

Explanation / Answer

(a) It is given that there will be X courses offered at a certain school next year. It is also given that X~P().

So, if there are four courses than by the additive property we know that X+X+X+X~P(+++)=P(4)

Therefore, the expected number of choices of four courses is 4.

(b) XAM and XPM are the number of morning and evening courses respectively.

It is given that the number of courses offered in the school is P().

So, the joint PMF of XAM and XPM is P( XAM=a and XPM=b) is given by: (e-a/a!)*(e-b/b!) for all a,b.

(c) It is given that X1, X2, X3, X4 are the number of 10am, 11:30am, 1pm and 2:30pm courses. The joint distribution is given by:

P(X1=a, X2=b, X3=c, X4=d) = (e-a/a!)*(e-b/b!)*(e-c/c!)*(e-d/d!)

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