Question

Why does the covariance of multinomial negative?Cov(X1,X2)= -np1p2

Answer 1

As we know for the multinomial distributionof the variable we have a fundamental constraint with the parameter n which has to be fulfilled each and every case as,

( or some cases it is as )

So we note that if keeping other variable constant if increases , then has to get decrease due to that constraint. And we also know that covariance is a form of correlation ( which is not normalized ) and the sign of it's value the relation between two variables (whether it is a direct or inverse relation).

So as   increases , then has to get decrease so it is an inverse relation between them and hence the covariance of multinomial is negative. ( which is true for any pair of multinomial variables)

Hence the answer................

Thank you.................