2.4 Multinomial
If \(X = (X_1, X_2, \cdots, X_k)\) are the counts of successful events in \(n\) identical and independent trials of success probabilities \(\pi = (\pi_1, \pi_2, \cdots, \pi_k)\), then \(X\) is a random variable with a multinomial distribution \(X \sim Mult(n,\pi)\)
\[f(x; n, \pi) = \frac{n!}{x_{1}! x_{2}! \cdots x_{k}!} \pi^{x_1} \pi^{x_2} \cdots \pi^{x_k} \hspace{1cm} x \in \{0, 1, ..., n \}, \hspace{2mm} \pi \in [0, 1]\]
with expected values vector \(E(X_j) = n\pi_j\) and covariance matrix
\[Var(X) = \begin{bmatrix}n\pi_{1}(1-\pi_{1}) & -n\pi_{1}\pi_{2} & \cdots & -n\pi_{1}\pi_{k}\\ -n\pi_{1}\pi_{2} & n\pi_{2}(1-\pi_{2}) & \cdots & -n\pi_{2}\pi_{k}\\ \vdots & \vdots & \ddots & \vdots\\ -n\pi_{1}\pi_{k} & -n\pi_{2}\pi_{k} & \cdots & n\pi_{k}(1-\pi_{k}) \end{bmatrix}\]
so \(Var(X_j) = n \pi_j (1 - \pi_j)\) and \(cov(X_j, X_k) = -n \pi_j \pi_k\).
The individual components of a multinomial random vector are binomial and have a binomial distribution, \(X_i = Bin(n, \pi_i)\). Binomial is a special case of multinomial for k = 2.
Suppose a city population is 20% black, 15% Hispanic, and 65% other. From a random sample of \(n = 12\) persons, what is the probability of 4 black and 8 other?
\[f(x;\pi) = \frac{12!}{4! 0! 8!} (0.20)^4 (0.15)^0 (0.65)^8 = 0.0252\]
Function dmultinom() calculates the multinomial probability.
## [1] 0.025To calculate the probability of <= 1 black, combine Hispanic and other, then sum the probability of black = 1 and black = 2.
\[f(x;\pi) = \frac{12!}{0! 12!} (0.20)^0 (0.80)^{12} + \frac{12!}{1! 11!} (0.20)^1 (0.80)^{11} = 0.2748\]
## [1] 0.27