University of California, Los Angeles Department of Statistics
Instructor: Nicolas Christou
Quiz 2
a. We discussed in class today the multinomial probability distribution and its joint moment generating func-
tion. Here is a note on the multinomial distribution: A sequence of n independent experiments is per-
formed and each experiment can result in one of r possible outcomes with probabilities p1, p2, . . . , pr with
ri=1 pi = 1. Let Xi be the number of the n experiments that result in outcome i, i = 1,2,…,r. Then,
P(X1 = x1,X2 = x2,…,Xr = xr) = n! px1px2 ···pxr. The joint moment generating function of the n1!n2!···nr! 1 2 r
multinomial distribution is given by MX(t) = (p1et1 +p2et2 +…+pretr)n. Use properties of joint moment generating functions to find the probability distribution of X1.
b. Refer to question (a). Use the joint moment generating function of the multinomial distribution and the theo- rem and corollary on handout #10, page 1 to find the mean and variance of X1.
c. Refer to question (a). Show that cov(Xi,Xj) = −npipj. Give an intuitive explanation of the negative sign.
d. Let X and Y be independent normal random variables, each with mean μ and standard deviation σ.
1. Consider the random quantities X + Y and X − Y . Find the moment generating function of X + Y and the moment generating function of X − Y .
2. Find the joint moment generating function of (X + Y, X − Y ).
3. Are X + Y and X − Y independent? Explain your answer using moment generating functions.
Statistics 100B
Answer the following questions:
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