Probability | Part IA, 2002

Define the indicator function IAI_{A} of an event AA.

Let IiI_{i} be the indicator function of the event Ai,1inA_{i}, 1 \leq i \leq n, and let N=1nIiN=\sum_{1}^{n} I_{i} be the number of values of ii such that AiA_{i} occurs. Show that E(N)=ipiE(N)=\sum_{i} p_{i} where pi=P(Ai)p_{i}=P\left(A_{i}\right), and find var(N)\operatorname{var}(N) in terms of the quantities pij=P(AiAj)p_{i j}=P\left(A_{i} \cap A_{j}\right).

Using Chebyshev's inequality or otherwise, show that

P(N=0)var(N){E(N)}2P(N=0) \leq \frac{\operatorname{var}(N)}{\{E(N)\}^{2}}

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