By Bryc W.

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9 and nd the easier sum n=0 1 , p . 2 Chebyshev-Markov inequality The following inequality is known as Markov's, or Chebyshev's inequality. Despite its simplicity it has numerous non-trivial applications, see eg. 1. 9 we have EU = 01 PrU x dx 0t PrU x dx tP jX j t. 8 Suppose U is uniform U 0; 1. Then PrU 1 , t 21t . t 21t . This means that 1. Is the above inequality sharp"? Graph both curves. 2. 9 sharp? That is, given t 0, is there an X 0 such that equality occurs? 28 CHAPTER 2.

In mathematical terms, random variable is a function X : ! IR with the property that sets f! ag are events in M for all a 2 IR. Recall that the last conditions means that we may talk about probabilities of events f! ag. Probabilities for a one-dimensional r. v. are determined by the cumulative distribution function F x = PrX x 1:12 The corresponding tail function Rx = 1 , F x = PrX x is sometimes called the reliability11 function. Cumulative distribution function can be used to express probabilities of intervals Pra X b = F b , F a.

10 L2-spaces Inequalities related to expected values are best stated in the geometric language of norms and normed spaces. We say that X 2 L2 , if X is square integrable, ie. EX 2 1. The L2 norm is q kX k2 = E jX j2: Notice that kX , EX k2 is just another notation for the standard deviation. Thus standard deviation is the L2 distance of X from a constant. We say that Xn converges to X in L2 , if kXn , X k2 ! 0 as n ! 1. We shall also use the phrase sequence Xn converges to X in mean-square. 1. Several useful inequalities are collected in the following4.

### Applied probability and stochastic processes by Bryc W.

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