Probability inequalities

CHAPTER 15 Probability inequalitiesWe already used several types of inequalities , and in this Chapter we give a more systematicdescription of the inequalities and bounds used in Probability and Boole s inequality, Bonferroni inequalitiesBoole s inequality(or theunion bound) states that for any at most countable collection ofevents, the Probability that at least one of the events happens is no greater than the sum ofthe probabilities of the events in the (Boole s inequality)Suppose(S,F,P)is a Probability space, andE1,E2,.. Fare events. ThenP( i=1Ei)6 i=1P(Ei). only give a proof for a finite collection of events, and we mathematicalinduction on the number of then= 1we see thatP(E1)6P(E1).Suppose that for somenand any collection of eventsE1,..,Enwe haveP(n i=1Ei)6n i=1P(Ei).Recall that by ( ) for any eventsAandBwe haveP(A B) =P(A) +P(B) P(A B).We apply it toA= ni=1 EiandB=En+1and using the associativity of the union n+1i=1Ei=A B, we get thatP(n+1 i=1Ei)=P(n i=1Ei) +P(En+1) P((n i=1Ei) En+1).

Here we revisit Chebyshev's inequality Proposition 14.1 we used previously. This results shows that the di erence between a random variable and its expectation is controlled by its variance. Informally we can say that it shows how far the random variable is from its mean on average. Proposition 15.4 (Chebyshev's inequality)

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  Name, Inequalities, Probability, Expectations, Probability inequalities

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