There will be X PoisA courses offered at a certain school ne

There will be X Pois(A) courses offered at a certain school next year. 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 X, and X pm be the number of morning and afternoon courses for next year am respectively (where \"morning\" means starting before noon). Find the joint PMF of Xam and X pm i.e., find P(X a, X, b) for all a, b. am pm

Solution

n1=number of courses before noon=2

p1=1/4

n2=number of courses after noon=2

p2=1/4

mean1=mean2=n1p1=n2p2=1/2

Let u=xam+xpm

let V=xpm

xam=u-v

f(u,v)=mean1^u-v*mean2^v-mean1-mean2/(u-v)!v!

Hence u is a poisson variabe with mean=mean1+mean2.