From Algorithms to Z-Scores - matloff - University of California, Davis

168 CHAPTER 8. MULTIVARIATE PMFS AND DENSITIES
We then use (8.95) to find the specified probability, which is:
6!
2!2!2! 0.202 0.16 2 0.64 2
8.5.1.3 Mean Vectors and Covariance Matrices in the Multinomial Family
(8.99)
Consider a multinomially distributed random vector X = (X1, ..., Xr) ′ , with n trials and category
probabilities pi. Let’s find its mean vector and covariance matrix.
First, note that the marginal distributions of the Xi are binomial! So,
So we know EX now:
EXi = npi and V ar(Xi) = npi(1 − pi) (8.100)
⎛
EX = ⎝
np1
...
npr
⎞
⎠ (8.101)
We also know the diagonal elements of Cov(X)—npi(1 − pi) is the i th diagonal element, i = 1,...,r.
But what about the rest? The derivation will follow in the footsteps of those of (3.104), but now
in a vector context. Prepare to use your indicator random variable, random vector and covariance
matrix skills! Also, this derivation will really build up your “probabilistic stamina level.” So, it’s
good for you! But now is the time to review (3.104), Section 3.6 and Section 7.3, before
continuing.
We’ll continue the notation of the last section. In order to keep on eye on the concrete, we’ll often
illustrate the notation with the die example above; there we rolled a die 8 times, and defined 6
categories (one dot, two dots, etc.). We were interested in probabilities involving the number of
trials that result in each of the 6 categories.
Define the random vector Ti to be the outcome of the i th trial. It is a vector of indicator random
variables, one for each of the r categories. In the die example, for instance, consider the second