Calculating Conditional Expectation for Continuous and Discrete Random Vectors

[dror_l]

Hi,

Let x,z continuous random vectors and n discrete random vector: n=[n1,n2,...].
I'm trying to find for instance, E_z|n3{ E_n|z(x)} = ?.

Thanks...

 

[EnumaElish]

Can you explain your notation? Do you have two questions, E_z|n3 = ? and E_n|z(x) = ?

 

[tacman]

Hello,

To calculate the conditional expectation for continuous and discrete random vectors, you will need to use the law of iterated expectations. This states that the expectation of a random variable is equal to the expectation of its conditional expectation, given another random variable. In other words, E[E(X|Y)] = E(X).

In your example, we have E_z|n3{ E_n|z(x)} = E[E_n|z(x)|n3]. This means that we first need to calculate the conditional expectation of E_n|z(x) given n3, and then take the overall expectation of this result.

To calculate E_n|z(x)|n3, we can use the conditional probability formula: P(A|B) = P(A and B)/P(B). In this case, we have E_n|z(x)|n3 = ∑x∫z P(z|x,n3)E_n|z(x)dz. This involves integrating over the possible values of z and summing over the possible values of n3.

Once we have calculated E_n|z(x)|n3, we can then take the overall expectation by summing over the possible values of n3, using the formula E(X) = ∑x P(x)X(x) for discrete random variables.

I hope this helps! Let me know if you have any further questions. Good luck with your calculations!