Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Stats Review Lecture 5 - Limit Theorems 07.25.12
Document related concepts
List of important publications in mathematics wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Fermat's Last Theorem wikipedia , lookup
Wiles's proof of Fermat's Last Theorem wikipedia , lookup
Nyquist–Shannon sampling theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Fundamental theorem of calculus wikipedia , lookup
Infinite monkey theorem wikipedia , lookup
Karhunen–Loève theorem wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Series (mathematics) wikipedia , lookup
Tweedie distribution wikipedia , lookup
Transcript
Chebyshev’s Inequality • Markov’s Inequality • Proposition 2.1. Chebyshev’s Inequality • Chebyshev’s Inequality: • Proposition 2.2. • Consider Example 2a Convergence in probability • A sequence of random variables, X1, X2, …, converges in probability to a random variable X if, for every e > 0, • or equivalently, The weak law of large numbers • Theorem 2.1. The weak law of large numbers • Proof: Almost Sure Convergence The Strong Law of Large Numbers • Theorem 4.1, p. 400 Convergence in distribution • A sequence of random variables, X1, X2, …, converges in distribution to a random variable X if • at all points x where FX(x) is continuous. • This really says that the CDFs converge Central Limit Theorem • Theorem 3.1. • For iid random variables Xi • Consider Examples 3b and 3c, p. 396 Central limit theorem for independent random variables • Theorem 3.2, p. 399. (a) The is uniformly bounded, meaning for some M, (b) and Jensen’s ineqality • Proposition 5.3, p. 409 • If f is convex • Consider Example 5f.
