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a) Let S = {1, 2, 3, 4) with each point carrying probability 1/4. Let A₁ = (1, 2), A2 = {1, 3), A3 = (1,4). Then any two of A1, A2, A3 are inde- pendent, but A1, A2, A3 are not independent. b) Let (A;, 1 ≤ i ≤ 5} be a measurable partition of 22 such that P(A1) = P(A2) = P(A3) 15/64, P(A4) = 1/64, P (A5) = 18/64. Define B = A₁ U A4, C = A2 U A4, D = A3 U A4. Check that P(BCD)= P(B)P (C) P (D) = but that B, C, D are not independent.

a) Let S = {1, 2, 3, 4) with each point carrying probability 1/4. Let A₁ = (1, 2), A2 = {1, 3), A3 = (1,4). Then any two of A1, A2, A3 are inde- pendent, but A1, A2, A3 are not independent. b) Let (A;, 1 ≤ i ≤ 5} be a measurable partition of 22 such that P(A1) = P(A2) = P(A3) 15/64, P(A4) = 1/64, P (A5) = 18/64. Define B = A₁ U A4, C = A2 U A4, D = A3 U A4. Check that P(BCD)= P(B)P (C) P (D) = but that B, C, D are not independent.

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 6E
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20. Counterexamples and examples: = a) Let S = {1, 2, 3, 4) with each point carrying probability 1/4. Let A₁ {1, 2}, A2 = {1,3), A3 {1, 4). Then any two of A1, A2, A3 are inde- pendent, but A1, A2, A3 are not independent. = b) Let (Ai, 1 ≤ i ≤ 5} be a measurable partition of 2 such that P(A₁) = P(A2) = P(A3) = 15/64, P(A4) = 1/64, P (A5) B = A1 U A4, C = A2 U A4, D = A3 U A4. Check that = 18/64. Define P(BCD)=P(B) P (C)P (D) but that B, C, D are not independent. c) Let X₁, X2 be independent random variables each assuming only the values +1 and -1 with probability 1/2. Are X1, X2, X1 X2 pairwise inde- pendent? Are X1, X2, X1 X2 an independent collection?
Transcribed Image Text:20. Counterexamples and examples: = a) Let S = {1, 2, 3, 4) with each point carrying probability 1/4. Let A₁ {1, 2}, A2 = {1,3), A3 {1, 4). Then any two of A1, A2, A3 are inde- pendent, but A1, A2, A3 are not independent. = b) Let (Ai, 1 ≤ i ≤ 5} be a measurable partition of 2 such that P(A₁) = P(A2) = P(A3) = 15/64, P(A4) = 1/64, P (A5) B = A1 U A4, C = A2 U A4, D = A3 U A4. Check that = 18/64. Define P(BCD)=P(B) P (C)P (D) but that B, C, D are not independent. c) Let X₁, X2 be independent random variables each assuming only the values +1 and -1 with probability 1/2. Are X1, X2, X1 X2 pairwise inde- pendent? Are X1, X2, X1 X2 an independent collection?
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