In: Statistics and Probability
Let N1 , N2 , N3 follow a trinomial distribution with parameters n, assume that n follows a Poisson distribution with parameter λ > 0. Also assume that, conditionally on N, the random variables N1, N2, N3 follow a trinomial distribution with N trials and category probabilities p1, p2, p3 with p1 + p2 + p3 = 1. Compute the covariance and correlation of (N1,N2)
The pmf of multinomial random variable
is defined as
where
This is sampling with replacement:
The Covariance
The variance
.
The correlation is
We see the correlation is independent of
For trinomial distribution,
. and conditional on
,
Since
,
.
The expected/ unconditional Covariance is
The conditional correlation is
.
The unconditional correlation is,