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

1, 2, 3, with respective probabilities p1, p2, p3,

3i

=1 pi = 1. Let Ni denote the number of trials that result in outcome i, and show that Cov(N1,N2) = −np1p2.

Also explain why it is intuitive that this covariance is negative. (Hint: For i =

1, . . . , n, let

Similarly, for j = 1, . . . , n, let

Argue that

Then use Proposition 4.7.2 and Theorem 4.7.4.)

Transcribed Image Text:

[1 if trial i results in outcome 1 X 0 if trial i does not result in outcome 1