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STAT 330
Christine Dupont

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STAT 330 Lecture, June 28, 2012 Last lecture: 1. Convergence in distribution (a) De▯nition (b) Examples 2. Convergence in probability (a) De▯nition of convergence in probability to a random variable (b) Theorem: convergence in probability implies convergence in distribution (c) De▯nition of convergence in probability to a constant (d) Theorem: convergence in probability to a constant = convergence in distribution to a degenerate distribution. 1 Example 2. (Example 1 Revisited.) Example 3. Suppose X ;▯▯▯;X are iid random variable with pdf 1 n ▯(x▯▯) f(x) = e ; x > ▯: Let X (1)= min(X ;:1:;X ). nhow that X (1)! p. 2 An important inequality: Markov Inequality Suppose X is a random variable. Then for any k > 0 and c > 0, we have k P(jXj ▯ c) ▯ E(jXj ) : c Proof. See Page 16 of the supplementary notes for the proof. A most commonly used result. 2 E(X ) P(jXj ▯ c) ▯ 2 : c Example 4. (Weak law of large numbers (WLLN)) Suppose X ;▯▯▯;X are 1ndepenn 2 dent random variable with the same mean ▯ and same variance ▯ . Show that n ▯ 1 X X = Xi! ▯p n i=1 3 Example 5. (Example 1 revisited fo(n) Recall that Xhas (n) 8 > > 0 x < 0
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