Final answer:
The correlation between counts xi and xj in a multinomial experiment is determined by applying Pearson's coefficient of correlation formula, using the corresponding variance and covariance of xi and xj, which ultimately reveals a negative association between the two.
Step-by-step explanation:
To show that the correlation between xi and xj in a multinomial experiment with counts x1, x2, ... , xk, trials n, and probabilities p1, p2, ... , pk, we need to refer to the formula for the coefficient of correlation developed by Karl Pearson and the properties of the multinomial distribution.
For a multinomial distribution, the expected value E(xi) is npi and E(xj) is npj. The covariance for xi and xj for i ≠ j is given by Cov(xi, xj) = -n * pi * pj. Since the variance Var(xi) of each xi can be calculated by npi(1-pi) and Var(xj) by npj(1-pj), we can find the correlation between xi and xj using Pearson's correlation coefficient formula:
corr(xi, xj) = Cov(xi, xj) / √(Var(xi) * Var(xj))
Substitute the values for covariance and variance into the formula to obtain the correlation between xi and xj. Since the covariance is negative, the correlation will be negative, indicating that as one of the variables increases, the other tends to decrease, which makes sense given that an increase in one category count in a multinomial experiment typically results in a decrease in another, due to the fixed number of trials.