[Expectation] For random variable X (with E(X) exists and finite), the Jensen's inequality provides the following conditions, Eg(X) ≤ (or <) g(EX), for g(X) is a concave (or strictly concave) function, ≥ (or >) g(EX), for g(X) is a convex (or strictly convex) function. Using this result, compare the following expectations: (a) E(X²) and (EX)²; (b) Elog(X) and log(EX); (c) E exp(X) and exp(EX).
[Expectation] For random variable X (with E(X) exists and finite), the Jensen's inequality provides the following conditions, Eg(X) ≤ (or <) g(EX), for g(X) is a concave (or strictly concave) function, ≥ (or >) g(EX), for g(X) is a convex (or strictly convex) function. Using this result, compare the following expectations: (a) E(X²) and (EX)²; (b) Elog(X) and log(EX); (c) E exp(X) and exp(EX).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Question
![[Expectation] For random variable X (with E(X) exists and finite), the Jensen's inequality provides
the following conditions,
Eg(X)
≤ (or <) g(EX), for g(X) is a concave (or strictly concave) function,
≥ (or >) g(EX), for g(X) is a convex (or strictly convex) function.
Using this result, compare the following expectations:
(a) E(X²) and (EX)²;
(b) Elog(X) and log(EX);
(c) E exp(X) and exp(EX).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F163cf53f-c0ce-4d67-85c3-58ee5a13e2c6%2Fd5e6043a-4afc-442a-9b25-a03523c0f07e%2F0frbggn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:[Expectation] For random variable X (with E(X) exists and finite), the Jensen's inequality provides
the following conditions,
Eg(X)
≤ (or <) g(EX), for g(X) is a concave (or strictly concave) function,
≥ (or >) g(EX), for g(X) is a convex (or strictly convex) function.
Using this result, compare the following expectations:
(a) E(X²) and (EX)²;
(b) Elog(X) and log(EX);
(c) E exp(X) and exp(EX).
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