NotesOct1

# NotesOct1 - Example Let X1 Xn be independent standard...

• Notes
• 10

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

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 .

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

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)
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.

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

Example Compute the characteristic function of the standard exponential random variable X .
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 ).

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

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

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

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

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern