Proving X=Y with Conditional Expectation

yavanna · 2009-11-30T08:19:56+00:00

How can I do this? Let X,Y r.v., \mathbb{E}(X|Y)=Y and \mathbb{E}(Y|X)=X. Proove that X=Y a.s.

Thread starter yavanna
Start date Nov 30, 2009
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Conditional Conditional expectation Expectation

In summary, conditional expectation is the expected value of a random variable, given the knowledge of some other related variable. It is calculated using the formula E(X|Y) = ∑x P(X=x|Y) and takes into account a specific condition or constraint. The difference between conditional expectation and unconditional expectation is that the latter does not have any conditions or constraints. Conditional expectation is commonly used in various fields to make predictions and decisions based on specific conditions. It can be negative if the random variable has a negative value, but the overall or unconditional expectation may still be positive.

Nov 30, 2009

yavanna

How can I do this?

Let X,Y r.v., [tex]\mathbb{E}(X|Y)=Y[/tex] and [tex]\mathbb{E}(Y|X)=X[/tex].
Proove that X=Y a.s.

Last edited: Nov 30, 2009

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Nov 30, 2009

bpet

yavanna said:

How can I do this?

Let X,Y r.v., [tex]\mathbb{E}(X|Y)=Y[/tex] and [tex]\mathbb{E}(Y|X)=X[/tex].
Proove that X=Y a.s.

could you give the definition of [tex]\mathbb{E}(X|Y)[/tex] then it might be easier

Related to Proving X=Y with Conditional Expectation

1. What is conditional expectation?

Conditional expectation is a statistical concept that refers to the expected value of a random variable, given the knowledge of some other related variable. In other words, it is the expected value of a variable when certain conditions or constraints are specified.

2. How is conditional expectation calculated?

The calculation of conditional expectation involves using the formula E(X|Y) = ∑x P(X=x|Y) where X and Y are random variables and P(X=x|Y) is the conditional probability of X given Y. This formula takes into account the probability of each possible outcome of X, given a specific value of Y, and calculates the expected value.

3. What is the difference between conditional expectation and unconditional expectation?

The unconditional expectation, also known as the overall or marginal expectation, is the expected value of a random variable without any conditions or constraints. On the other hand, conditional expectation takes into account a specific condition or constraint and calculates the expected value based on that condition.

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

Conditional expectation is commonly used in various fields, such as finance, economics, and engineering, to make predictions and decisions based on specific conditions. For example, in finance, it can be used to estimate the expected return on investment given certain market conditions or economic factors.

5. Can conditional expectation be negative?

Yes, conditional expectation can be negative if the random variable has a negative value. This means that the expected value of the variable under a specific condition is negative. However, the overall or unconditional expectation may still be positive, depending on the probability distribution of the variable.