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lecturenotesweek15

# lecturenotesweek15 - ISyE 2027 Probability with...

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ISyE 2027 Probability with Applications Polly B. He Fall10, Week 15 Reading: Section 13.1-3, 14.1-2 The Law of Large Numbers Intuitively, we view the probability of a certain outcome as the proportion of times that this outcome occurs when the experiment is repeated many times over a long period of time. Mathematically, we deﬁne probability as the value, say, of a p.m.f. for a random variable representing the outcome. How can we be sure that these two are consistent? Averages of a Sequence of Random Variables Suppose X 1 ,X 2 ,...,X n are n independent, identically distributed random variables (iden- tically distributed means these random variables have the same distribution). Then the average of this sequence is ¯ X n = X 1 + X 2 + ··· + X n n If ¯ X n is the average of n independent random variables with the same expectation µ and variance σ 2 , then E[ ¯ X n ] = µ and Var( ¯ X n ) = σ 2 n Chebyshev’s Inequality The following inequality provides a bound for the probability that the random variable

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lecturenotesweek15 - ISyE 2027 Probability with...

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