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Example 3.1 This is a slight variation of Example 2.5.1. Let, for a > 0, X1, X2, ... be independent random variables such that 1 1 P(X = 0) 1- na and P(Xn) = n≥ 1. na' The following statements hold: Xn 0 as 818 even without independence, Xn as 0 as n→∞ iff a 1, Xn 0 as Xn → 0 as 818 818 iff a 1, iff ar. Convergence in probability is a consequence of the fact that 1 P(X) = P(X = n) = →0 as n→ ∞. na The complete and almost sure facts follow from the Borel-Cantelli lemmas, since ∞ <+∞0 when a 1, ΣP (|xn| > €) = +∞ when a ≤1. n=1 As for mean convergence, EX=0" (1- (1 1 +n". na na → 0, for ra.

Example 3.1 This is a slight variation of Example 2.5.1. Let, for a > 0, X1, X2, ... be independent random variables such that 1 1 P(X = 0) 1- na and P(Xn) = n≥ 1. na' The following statements hold: Xn 0 as 818 even without independence, Xn as 0 as n→∞ iff a 1, Xn 0 as Xn → 0 as 818 818 iff a 1, iff ar. Convergence in probability is a consequence of the fact that 1 P(X) = P(X = n) = →0 as n→ ∞. na The complete and almost sure facts follow from the Borel-Cantelli lemmas, since ∞ <+∞0 when a 1, ΣP (|xn| > €) = +∞ when a ≤1. n=1 As for mean convergence, EX=0" (1- (1 1 +n". na na → 0, for ra.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
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Example 3.1 This is a slight variation of Example 2.5.1. Let, for a > 0, X1, X2, ... be independent random variables such that 1 1 P(X = 0) 1- na and P(Xn) = n≥ 1. na' The following statements hold: Xn 0 as 818 even without independence, Xn as 0 as n→∞ iff a 1, Xn 0 as Xn → 0 as 818 818 iff a 1, iff ar. Convergence in probability is a consequence of the fact that 1 P(X) = P(X = n) = →0 as n→ ∞. na The complete and almost sure facts follow from the Borel-Cantelli lemmas, since ∞ <+∞0 when a 1, ΣP (|xn| > €) = +∞ when a ≤1. n=1 As for mean convergence, EX=0" (1- (1 1 +n". na na → 0, for r<a, = n-a = 1, for r = a, as no. 8+← for r>a.
Transcribed Image Text:Example 3.1 This is a slight variation of Example 2.5.1. Let, for a > 0, X1, X2, ... be independent random variables such that 1 1 P(X = 0) 1- na and P(Xn) = n≥ 1. na' The following statements hold: Xn 0 as 818 even without independence, Xn as 0 as n→∞ iff a 1, Xn 0 as Xn → 0 as 818 818 iff a 1, iff ar. Convergence in probability is a consequence of the fact that 1 P(X) = P(X = n) = →0 as n→ ∞. na The complete and almost sure facts follow from the Borel-Cantelli lemmas, since ∞ <+∞0 when a 1, ΣP (|xn| > €) = +∞ when a ≤1. n=1 As for mean convergence, EX=0" (1- (1 1 +n". na na → 0, for r<a, = n-a = 1, for r = a, as no. 8+← for r>a.
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