Limits theorem - Chapter 5 Limits Theorem The most...

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Chapter 5 Limits Theorem The most important theoretical results in probability theory are limit theorems. Of these, the most important are Laws of large numbers and Central limit theorems . Definition 5.0.119 ( Markov’s inequality ) . If X is a random variable that takes only non- negative values, then, for any value a > 0, P ( X a ) E [ X ] a . Definition 5.0.120 ( Chebyshev’s inequality ) . If X is a random variable with finite mean μ and variance σ 2 , then, for any value k > 0, P ( | X - μ | ≥ k ) σ 2 k 2 . Remark 5.0.121 . The importance of Markov’s and Chebyshev’s inequalities is that they enable us to derive bounds on probabilities when only the mean, or both the mean and the variance are known. Example 5.0.122. If V ar ( X ) = 0, then P ( X = E [ X ]) = 1. In other words, the only random variables having variances equal to 0 are those which are constant with probability 1. Definition 5.0.123 ( Sample mean ) . Let X be a random variable for which the mean, E [ X ] = μ , is unknown. Let X 1 , X 2 , .... denotes n independent repeated measurements of X; that is, the X 0 j s are independent, identically distributed (iid) random variables with the same pdf as X. The sample mean of the sequence is M n = 1 n n X i =1 X j Remark 5.0.124 . The expected value and variance of the sample mean are μ and σ 2 n respectively. 18

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CHAPTER 5. LIMITS THEOREM Example 5.0.125. Let X 1 , X 2 , .... denotes n independent, identically distributed (iid) random variables, each with mean μ and variance σ 2 . If M n
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