The means and variances of the individual variables in a multinomial distribution follow directly from their marginal binomial distributions.
Means and variances
If \((X_1, \dots, X_g)\) have a \(\MultinomDistn(n, \pi_1, \dots, \pi_g)\) distribution,
\[ E[X_i] \;=\; n\pi_i \spaced{and} \Var(X_i) \;=\; n\pi_i(1 - \pi_i) \qquad \text{for }i=1,\dots,g \]The covariance between any two variables in a multinomial distribution is a little harder to obtain.
Covariances
If \((X_1, \dots, X_g)\) have a \(\MultinomDistn(n, \pi_1, \dots, \pi_g)\) distribution,
\[ \Covar(X_i, X_j) \;=\; -n\pi_i\pi_j \qquad \text{if }i \ne j \](Proved in full version)
The correlation between any two of the multinomial variables is:
Correlation coefficients
If \((X_1, \dots, X_g)\) have a \(\MultinomDistn(n, \pi_1, \dots, \pi_g)\) distribution,
\[ \Corr(X_i, X_j) \;=\; -\sqrt{\frac{\pi_i\pi_j}{(1 - \pi_i)(1 - \pi_j)}} \](Proved in full version)
The correlation between two multinomial variables only depends on the probabilities \(\pi_i\) and \(\pi_j\), not on the sample size, \(n\).