20.21. Suppose that the sequence (X) is fundamental in probability in the sense that for e positive there exists an N such that P[IX-X|>e] N. (a) Prove there is a subsequence (X) and a random variable X such that lim XX with probability 1. Hint: Choose increasing n such that P[IX-X|>2-k]<2-k for m, n ≥n. Analyze P[IX-Xn₂>2-k]. (b) Show that X,₁ →p X.
20.21. Suppose that the sequence (X) is fundamental in probability in the sense that for e positive there exists an N such that P[IX-X|>e] N. (a) Prove there is a subsequence (X) and a random variable X such that lim XX with probability 1. Hint: Choose increasing n such that P[IX-X|>2-k]<2-k for m, n ≥n. Analyze P[IX-Xn₂>2-k]. (b) Show that X,₁ →p X.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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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![20.21. Suppose that the sequence (X) is fundamental in probability in the sense that
for e positive there exists an N such that P[IX-X|>e] <e for m, n > N.
(a) Prove there is a subsequence (X) and a random variable X such that
lim XX with probability 1. Hint: Choose increasing n such that
P[IX-X₁>2-k] <2-k for m, n ≥n. Analyze P[X-Xl>2-k].
→P
X.
(b) Show that Xn](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3d557df-911d-4c1c-951f-1faffc0aa610%2F3a912d16-6e24-471a-8085-7b12369b575f%2Fbodny6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:20.21. Suppose that the sequence (X) is fundamental in probability in the sense that
for e positive there exists an N such that P[IX-X|>e] <e for m, n > N.
(a) Prove there is a subsequence (X) and a random variable X such that
lim XX with probability 1. Hint: Choose increasing n such that
P[IX-X₁>2-k] <2-k for m, n ≥n. Analyze P[X-Xl>2-k].
→P
X.
(b) Show that Xn
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