14. In a sequence of independent Bernoulli random variables (Xn, n ≥ 1) with P[Xn = 1] = p = 1 - P[X, = 0], let An be the event that a run of n consecutive 1's occurs between the 2n and 2+1st trial. If p≥ 1/2, then there is probability 1 that infinitely many An occur. Hint: Prove something like P(An) ≥ 1-(1-p")2" /2n >1-e-(2p)" /2n
14. In a sequence of independent Bernoulli random variables (Xn, n ≥ 1) with P[Xn = 1] = p = 1 - P[X, = 0], let An be the event that a run of n consecutive 1's occurs between the 2n and 2+1st trial. If p≥ 1/2, then there is probability 1 that infinitely many An occur. Hint: Prove something like P(An) ≥ 1-(1-p")2" /2n >1-e-(2p)" /2n
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![14. In a sequence of independent Bernoulli random variables (Xn, n ≥ 1) with
P[X₂ = 1] = p = 1 - P[X₂ = 0],
let An be the event that a run of n consecutive 1's occurs between the 2"
and 2n+1st trial. If p≥ 1/2, then there is probability 1 that infinitely many
An occur.
Hint: Prove something like
P(An) ≥ 1-(1-p")2" /2n >1-e-(2p)" /2n](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41a08612-16ea-42a8-bd61-0edbeea1fc19%2F0e1228b6-ea63-4f85-a26a-2cd5cd8c8774%2Fxgazsjv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:14. In a sequence of independent Bernoulli random variables (Xn, n ≥ 1) with
P[X₂ = 1] = p = 1 - P[X₂ = 0],
let An be the event that a run of n consecutive 1's occurs between the 2"
and 2n+1st trial. If p≥ 1/2, then there is probability 1 that infinitely many
An occur.
Hint: Prove something like
P(An) ≥ 1-(1-p")2" /2n >1-e-(2p)" /2n
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