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

To know:Distribution of the sum of multinomial random variables.
Covariance of multinomial random variables,
$COV\left({N}_{i},{N}_{j}\right)=-m{P}_{i}{P}_{j},$
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berggansS
${N}_{i}+{N}_{j}:$ sum of indicator variables.
For obtaining the result, we could have used
$Var\left({N}_{i}+{N}_{j}\right)=Var\left({N}_{i}\right)+Var\left({N}_{j}\right)+2Cov\left({N}_{i},{N}_{j}\right)$
We have
$Var\left({N}_{i}+{N}_{j}\right)=Var\left({N}_{i}\right)+Var\left({N}_{j}\right)+2Cov\left({N}_{i},{N}_{j}\right)$
Such that
$Var\left({N}_{i}+{N}_{j}\right)=m\left({P}_{i}+{P}_{j}\right)\left(1-{P}_{i}-{P}_{j}\right)$
Rewrite the above
$Var\left({N}_{i}+{N}_{j}\right)=m\left({P}_{i}+{P}_{j}\right)\left(1-\left({P}_{i}+{P}_{j}\right)\right)$
Thus,
With parameters m and $\left({P}_{i}+{P}_{j}\right)$,
The distribution of the sum of multinomial random variables is Binomial.