2 Law of Large Numbers Define x a random variable with E[x] = µ, V[x] = o² <∞, and a = |x| > 0. You can write the unconditional expectation as E[a] = E[ala c] × P[a >c] X (1) 1. Prove E[ala
2 Law of Large Numbers Define x a random variable with E[x] = µ, V[x] = o² <∞, and a = |x| > 0. You can write the unconditional expectation as E[a] = E[ala c] × P[a >c] X (1) 1. Prove E[ala
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![2 Law of Large Numbers
Define x a random variable with E[x] = µ‚V[x] = o² < ∞, and a = = |x| > 0. You can write the unconditional expectation as
E[a] = E[a|a < c] × P[a < c] + E[a]a ≥ c] × P[a ≥ c]
(1)
1. Prove E[ala <c] ≥ 0
2. Prove E[ala ≥c] ≥ c
3. Using these two results, prove P[a ≥ c] ≤ E[a]
с
4. Using the previous result, prove P[|x − µ| ≥ ko] ≤
k2
5. Using the results in part 1, prove P[|yo - Ho| ≥ k-
02
for k> 0
=] ≤ k- for k > 0
6. What is the limit of P[|7o – Hol ≥
as No → ∞?
You have just proved the law of large numbers (LLN). The LLN states the sample average will grow ever closer to the
expected value. In practice, with large samples, you can interchange the expected value and sample mean.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79c29685-e383-4a24-86b7-339f6b2dc05d%2Fd07d89ec-9b95-4b90-a21a-91513cca8f95%2Flwss5vo_processed.png&w=3840&q=75)
Transcribed Image Text:2 Law of Large Numbers
Define x a random variable with E[x] = µ‚V[x] = o² < ∞, and a = = |x| > 0. You can write the unconditional expectation as
E[a] = E[a|a < c] × P[a < c] + E[a]a ≥ c] × P[a ≥ c]
(1)
1. Prove E[ala <c] ≥ 0
2. Prove E[ala ≥c] ≥ c
3. Using these two results, prove P[a ≥ c] ≤ E[a]
с
4. Using the previous result, prove P[|x − µ| ≥ ko] ≤
k2
5. Using the results in part 1, prove P[|yo - Ho| ≥ k-
02
for k> 0
=] ≤ k- for k > 0
6. What is the limit of P[|7o – Hol ≥
as No → ∞?
You have just proved the law of large numbers (LLN). The LLN states the sample average will grow ever closer to the
expected value. In practice, with large samples, you can interchange the expected value and sample mean.
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