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

pedzenekO 2021-01-19 Answered
To know:Distribution of the sum of multinomial random variables.
Covariance of multinomial random variables,
COV(Ni,Nj)=mPiPj,
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berggansS
Answered 2021-01-20 Author has 91 answers
Ni+Nj: sum of indicator variables.
For obtaining the result, we could have used
Var(Ni+Nj)=Var(Ni)+Var(Nj)+2Cov(Ni,Nj)
We have
Var(Ni+Nj)=Var(Ni)+Var(Nj)+2Cov(Ni,Nj)
Such that
Var(Ni+Nj)=m(Pi+Pj)(1PiPj)
Rewrite the above
Var(Ni+Nj)=m(Pi+Pj)(1(Pi+Pj))
Thus,
With parameters m and (Pi+Pj),
The distribution of the sum of multinomial random variables is Binomial.
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