Paper 2, Section II, F

Probability | Part IA, 2019

Let A1,A2,,AnA_{1}, A_{2}, \ldots, A_{n} be events in some probability space. Let XX be the number of AiA_{i} that occur (so XX is a random variable). Show that

E(X)=i=1nP(Ai)\mathbb{E}(X)=\sum_{i=1}^{n} \mathbb{P}\left(A_{i}\right)


Var(X)=i=1nj=1n(P(AiAj)P(Ai)P(Aj))\operatorname{Var}(X)=\sum_{i=1}^{n} \sum_{j=1}^{n}\left(\mathbb{P}\left(A_{i} \cap A_{j}\right)-\mathbb{P}\left(A_{i}\right) \mathbb{P}\left(A_{j}\right)\right)

[Hint: Write X=i=1nXiX=\sum_{i=1}^{n} X_{i} where Xi={1 if Ai occurs 0 if not X_{i}=\left\{\begin{array}{ll}1 & \text { if } A_{i} \text { occurs } \\ 0 & \text { if not }\end{array}\right..]

A collection of nn lightbulbs are arranged in a circle. Each bulb is on independently with probability pp. Let XX be the number of bulbs such that both that bulb and the next bulb clockwise are on. Find E(X)\mathbb{E}(X) and Var(X)\operatorname{Var}(X).

Let BB be the event that there is at least one pair of adjacent bulbs that are both on.

Use Markov's inequality to show that if p=n0.6p=n^{-0.6} then P(B)0\mathbb{P}(B) \rightarrow 0 as nn \rightarrow \infty.

Use Chebychev's inequality to show that if p=n0.4p=n^{-0.4} then P(B)1\mathbb{P}(B) \rightarrow 1 as nn \rightarrow \infty.

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