How to Prove E[Y|F0]=Y When Y is F0-Measurable?

In summary, the conversation revolves around understanding conditional expectation and proving that if Y is a measurable F0 random variable and E[|Y|]<∞, then E[Y|F0]=Y. The definition of conditional expected value and its properties are also mentioned. The person asking for help is advised to either share their attempts or consult a textbook for the solution.
  • #1
umutk
2
0
I need help about conditional expectation for my research. I get stucked on this point. Could anyone show me how to prove that:
"Let E[|Y|]<∞. By checking that Definition is satisfied, show that if Y is measurable F0, then E[Y|F0]=Y."

Def: Let Y be a random variable defined on an underlying probability space([tex]\Omega[/tex],F,P) and satisfying E[|Y|]<∞. Let F0 be a sub-[tex]\sigma[/tex]-algebra of F. The conditional expected value of Y given F0,denoted E[Y|F0],is an F0-measurable random variable that also satisfies:E[IFY]=E[IFE[Y|F0]] for all F [tex]\in[/tex] F0

Note that: Red Fs are sets, but black Fs are sigma-algebras.

I appreciate any response.
 
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  • #2
anybody??
 
  • #3
Welcome to PF.
Are you really trying to tell us that is research? To me it sounds like an exercise from a measure theoretic probability course.
I suggest you show what you have tried so far or look up the answer in a textbook.
 

FAQ: How to Prove E[Y|F0]=Y When Y is F0-Measurable?

1. What is conditional expectation?

Conditional expectation is a concept in probability and statistics that represents the expected value of a random variable given that another random variable has taken on a certain value or falls within a certain range. It is calculated by taking the expected value of the conditional probability distribution.

2. How is conditional expectation different from regular expectation?

Regular expectation, also known as unconditional expectation, is the expected value of a random variable without any conditions. Conditional expectation takes into account an additional variable and calculates the expected value based on that condition.

3. What is the formula for conditional expectation?

The formula for conditional expectation is E(X|Y) = ∑x P(X=x|Y) * x, where E(X|Y) represents the conditional expectation of variable X given variable Y, P(X=x|Y) is the conditional probability of X taking on value x given that Y has occurred, and x is the value of X.

4. How is conditional expectation used in real-life applications?

Conditional expectation is commonly used in finance, economics, and other fields to make predictions and decisions based on data with multiple variables. It can be used to analyze risk, make investment decisions, and predict future outcomes.

5. What are some common properties of conditional expectation?

Some common properties of conditional expectation include linearity, which means that E(aX + bY|Z) = aE(X|Z) + bE(Y|Z) where a and b are constants, and the law of iterated expectations, which states that E(E(X|Y)) = E(X). Conditional expectation also follows the law of total probability and the law of total variance.

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