2. A certain 400-level course has 5 freshmen, 10 sophomores, 15 juniors, and 20 seniors. Let
us take a random sample of 10 people with or without replacement. Let X be the number of
freshmen and sophomores (total), Y be the number of juniors, and Z be the number of seniors
in a random sample of size 10.
(a) (2 points) Find the joint PMF of X, Y, Z, $P(X = x, Y = y, Z = z)$, when the sampling was
without replacement.
Hint: This will be a generalization of Hypergeometric PMF with three groups. One example was
given in Packet 8; the number of red marbles, that of white marbles, and that of black marbles
when two marbles are selected out of nine.
(b) (2 points) Find the joint PMF of X, Y, Z, $P(X = x, Y = y, Z = z)$, when the sampling was
with replacement.
Hint: In sampling with replacement, the probability of selecting a freshman remains the same
across the sampling procedure. So it must be a generalization of Binomial PMF with three
categories (groups). That is, we can think of this question as throwing 10 balls into 3 boxes.
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(c) (2 points) Find $E(Y+Z)$ and $Var(Y+Z)$, when the sampling was with replacement.
(d) (2 points) Find $Corr(Y, Z)$, when the sampling was with replacement.
