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
asked 2021-01-19
To know:Distribution of the sum of multinomial random variables.
Covariance of multinomial random variables,
\(COV(N_{i},N_{j})=-mP_{i}P_{j},\)

Answers (1)

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.
0

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