STAT330 Lecture Notes - Degenerate Distribution, Independent And Identically Distributed Random Variables, Random Variable

• School: University of Waterloo
• Department: Statistics
• Course: STAT330
• Professor: Christine Dupont

Document Summary

Suppose x1, , xn are iid random variable with pdf f (x) = e (x ), x > . Then for any k > 0 and c > 0, we have. See page 16 of the supplementary notes for the proof. P (|x| c) e(x 2) c2. Example 4. (weak law of large numbers (wlln)) suppose x1, , xn are indepen- dent random variable with the same mean and same variance 2. Recall that x(n) has cdf : p (x(n) x) = Example 5. (example 1 revisited for x(n)) x < 0. Let x1, x2, , xn be i. i. d r. v. s with e(xi) = and (cid:80)n. Let xn = 1 n( xn ) i=1 xi. A useful result for proving central limit theorem. Let x1, x2, , xn, be a se- quence of r. v. s such that xn has m. g. f mn(t) and let x be a r. v. with m. g. f m (t).