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

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
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 .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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)
Background image of page 2
The characteristic function ψ X ( t ) of a random variable X is defined by ψ X ( t ) = ψ ( t ) = E ± e itX ² , 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.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Example Compute the characteristic function of the standard exponential random variable X .
Background image of page 4
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 ). Recall that the sample mean of the first
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
This is the end of the preview. Sign up to access the rest of the document.

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 → ∞ ....
View Full Document

This note was uploaded on 12/03/2010 for the course OR&IE 3500 taught by Professor Samorodnitsky during the Fall '10 term at Cornell University (Engineering School).

Page1 / 10

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

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online