# To know:Distribution of the sum of multinomial random variables. Covariance of multinomial random variables, COV(N_{i},N_{j})=-mP_{i}P_{j},

Question
Random variables
To know:Distribution of the sum of multinomial random variables.
Covariance of multinomial random variables,
$$COV(N_{i},N_{j})=-mP_{i}P_{j},$$

2021-01-20
$$N_{i} + N_{j}:$$ sum of indicator variables.
For obtaining the result, we could have used
$$Var (N_{i} + N_{j}) = Var (N_{i}) + Var (N_{j}) + 2Cov(N_{i},N_{j})$$
We have
$$Var (N_{i} + N_{j}) = Var (N_{i}) + Var (N_{j}) + 2Cov (N_{i}, N_{j})$$
Such that
$$Var (N_{i} + N_{j}) = m(P_{i} + P_{j}) (1-P_{i}-P_{j})$$
Rewrite the above
$$Var(N_{i}+N_{j})=m(P_{i}+P_{j})(1-(P_{i}+P_{j}))$$
Thus,
With parameters m and $$(P_{i}+P_{j})$$,
The distribution of the sum of multinomial random variables is Binomial.

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