Stat120C Homework 6
Instructor:
Problem 1: Let Y = (Y1, Y2, Y3) follows a multinomial distribution with n trials and probabilities p =
(p1, p2, p3). Noet that the sum of pi's is 1, i.e., p1 + p2 + p3 = 1. The probability mass function (pmf) is
Pr(Y = (y1, y2, y3)) = n!
y1!y2!y3!py1
1 py2
2 py3
3
Here y1, y2, y3 are nonnegative integers that satisfy y1 + y2 + y3 = n.
(a) Show that Y1 follows Binomial(n, p1) by showing that
Pr(Y1 = y1) =
X
y2,y3
P(Y1 = y1, Y2 = y2, Y3 = y3) = n!
y1!(n - y1)!py1
1 (1 - p1)n-y1
Hint: the Binomial theorem is useful:
(a + b)n =
Xn
x=0
n!
x!(n - x)!axbn-x
(b) Prove that Cov(Y1, Y2) = -np1p2,Cov(Y1, Y3) = -np1p3,Cov(Y2, Y3) = -np2p3.
Hint: E[Y1Y2] =
P
y1,y2,y3 y1y2Pr(Y1 = y1, Y2 = y2, Y3 = y3). Show that it equals n(n - 1)p1p2. The
trinomial theorem is useful: (a + b + c)n =
P
x+y+z=n
n!
x!y!z!axbycz.