Consider n independent trials each of which results in any o

Consider n independent trials, each of which results in any of the outcomes i i = 123

with respective probabilities p1p2p3 such that p1 + p2 + p + 3 = 1. Let N_i denote the number

of trials that result in outcome i, and show that Cov(N₁N₂) = -np₁p₂. Also explain why it is

intuitive that this covariance is negative.

Solution

p1 = 1 / P2+P2

cov (n1,n2) = p1 + p2 + p3 / n * n^2 /p^3

cov (n1,n2) = - p1*´p2*n

is negative because the relationship between the variables in inverse proportional