NotesOct1 - n ob-servations is defined by ¯ X n = ¯ X =...

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Example Let X 1 , . . . , X n be independent stan- dard exponential random variables, and Y = X 1 + . . . + X n . Compute the Laplace transform of Y .
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The characteristic function of a random vari- able X is used when the moment generating function and the Laplace transform are not de- fined. If t > 0 then e tX becomes very large if X is very large If t < 0 then e tX becomes very large if X is very small (negative)
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The characteristic function ψ X ( t ) of a random variable X is defined by ψ X ( t ) = ψ ( t ) = E e itX , i = - 1 . Since | e ia | = 1 for any real a , the characteristic function of a random variable is always defined. Characteristic functions have similar properties and applications to those of moment generat- ing functions and Laplace transforms.
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Example Compute the characteristic function of the standard exponential random variable X .
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Law of Large Numbers Suppose X 1 , X 2 , . . . be independent and iden- tically distributed (iid) random variables with mean μ = EX j and variance σ 2 = Var( X j ).
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Unformatted text preview: n ob-servations is defined by ¯ X n = ¯ X = X 1 + X 2 + ... + X n n , n = 1 , 2 ,... . Recall that E ( ¯ X n ) = μ, and Var ( ¯ X n ) = σ 2 n , n = 1 , 2 ,... . Conclusion : The sample mean becomes more and more concentrated around the (popula-tion) true mean μ . The Law of Large Numbers : The sample mean of a sequence of iid random variables with mean μ , converges to this true mean as the sample size n grows. ¯ X n → μ as n → ∞ . 1 . Markov’s inequality For any nonnegative random variable X and a number a > P ( X > a ) ≤ EX a . 2 . Chebyshev’s inequality For any random variable X with mean μ , and number h > P ( | X-μ | > h ) ≤ Var( X ) h 2 . 3 . The Law of Large Numbers For any ± > P (± ± ¯ X n-μ ± ± > ± ) → 0 as n → ∞ ....
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