Relationship between two random variables having same expectation

AI Thread Summary
The discussion centers on the relationship between two functions, f(X) and g(X), of the same random variable X, given that their expected values are equal, E_X[f(X)] = E_X[g(X)] = a. One participant proposes that f(X) can be expressed as f(X) = g(X) + h(X), where the expectation of h(X) is zero, E_X[h(X)] = 0. The question raised is whether this relationship is both necessary and sufficient. The conversation seeks clarification on the implications of equal expectations for the functions involved. Understanding this relationship is crucial for deeper insights into the behavior of random variables and their transformations.
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Homework Statement



Say, it is known that
E_X[f(X)] = E_X[g(X)] = a where f(X) and g(X) are two functions of the same random variable X. What is the relationship between f(X) and g(X)?

Homework Equations





The Attempt at a Solution



My answer is f(X) = g(X) + h(X) where E_X[h(X)] = 0.

This relationship is apparent by taking expectation in both sides. But, is it necessarily and sufficiently true?
 
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