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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.