Questions

Housing Day

Suppose Harvard College is conducting its housing lottery. For simplicity's sake, we'll say that there are 1200 Freshmen that will be randomly assigned to 12 houses. Let X1,X2,,X12X_1, X_2, \ldots, X_{12} count how many students are place in Pforzheimer (X1X_1), all the way to Eliot (X12X_{12}) (organized by best house to worst).

Are X1X_1 and X2X_2 independent?

What is the joint distribution of X1,X2,,X12X_1, X_2, \ldots, X_{12}?

What is the marginal distribution of X1X_1, the number of students who are placed into Pforzheimer House, and the joint distribution of X1X_1 and 1200X11200 - X_1?

What is the conditional distribution of X1X_1 given X10+X11+X12=450X_{10} + X_{11} + X_{12} = 450?

Jelly Beans

I have a jar of 30 jellybeans: 10 red, 8 green, 12 blue. I draw a sample of 12 jellybeans without replacement. Let XX be the number of red jellybeans in the sample, YY the number of green jellybeans.

Find Cov(X,Y)Cov(X, Y).

Stat Courses

Let XX 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 a track in quant finance. Suppose that each statistics major chooses randomly which of these two tracks to follow, independently, with probability pp of choosing the general track. Let YY be the number of statistics majors who choose the general track, and ZZ be the number of statistics majors who choose the quantitative finance track.

Suppose that XPois(λ)X \sim \text{Pois}(\lambda). Find the correlation between XX and YY.

Let nn be the size of the Class of 2030, where nn is a known constant. For this part and the next, instead of assuming that XX is Poisson, assume that each of the nn students chooses to be a statistics major with probability rr, independently. Find the joint distributions of YY, ZZ, and the number of non-statistics majors, and their marginal distributions.

Continuing as in the previous part, find the correlation between XX and YY.

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