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In: Statistics and Probability

Let N1 , N2 , N3 follow a trinomial distribution with parameters n, assume that n...

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)

Solutions

Expert Solution

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,

orchestra answered 1 year ago
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