Consider n independent trials, each of which results in any of the outcomes
Consider n independent trials, each of which results in any of the outcomes i, i =
 1, 2, 3, with respective probabilities p1, p2,  p3, ),3          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
Xi =
Â
f1 Â Â if trial i results in outcome 1
0 Â Â if trial i does not result in outcome 1
Similarly, for j = 1, ... , n, let
f1 Â Â if trial j results in outcome 2
Yj =
0 Â Â if trial j does not result in outcome 2
Argue that
n               n
N1 = )' Xi , Â Â N2 = )' Yj
i=1
Then use Proposition 4.7.2 and Theorem 4.7.4.)
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Answers
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Posted 07 May 2017 10:05 AM
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