Parameterizing conditional expectations (Gaussian case)

In summary, the conversation discusses the regression coefficients of three jointly normally distributed variables. The derivation provided is correct and involves the Tower Law and the linearity of the expectation operator.
  • #1
estebanox
26
0
Consider three jointly normally distributed random variables X,Y and Z.

I know that in the Gaussian case E[Z | X,=x Y=y]=xßZX;Y +yßZY;X
where ßZX;Y notes the regression coefficient of Z on X conditional on Y (and ßZY;Xis analogously defined).

Is the following derivation correct?

E[Z| X>x, Y<y] = E [ E[Z | X,Y] | X>x, Y<y]
=ßZX;Y E[X | X>x, Y<y]+ßZY;X E[Y |X>x, Y<y]
 
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  • #2
Yes it is correct.
The first equality is an application of the Tower Law.
The second equality actually comprises two steps:
1. substitute your formula for E[Z | X,Y] from above for the inner expectation
2. Use the linearity of the expectation operator and the fact that the betas are constants to get the result.
 
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  • #3
andrewkirk said:
Yes it is correct.
The first equality is an application of the Tower Law.
The second equality actually comprises two steps:
1. substitute your formula for E[Z | X,Y] from above for the inner expectation
2. Use the linearity of the expectation operator and the fact that the betas are constants to get the result.
Great! Thanks a lot for spelling it out
 

Related to Parameterizing conditional expectations (Gaussian case)

What is parameterizing conditional expectations?

Parameterizing conditional expectations is a statistical method used to estimate the relationship between a dependent and independent variable in a dataset. It involves fitting a model to the data and using the parameters of the model to predict the value of the dependent variable given a certain value of the independent variable.

What is the Gaussian case in parameterizing conditional expectations?

The Gaussian case refers to a specific type of distribution, the Gaussian or normal distribution, that is commonly used in statistical modeling. In this case, the conditional expectations are parameterized using the parameters of the Gaussian distribution, such as the mean and standard deviation.

What are the benefits of using parameterizing conditional expectations in the Gaussian case?

Using parameterizing conditional expectations in the Gaussian case allows for a more accurate estimation of the relationship between variables compared to other methods. It also allows for the incorporation of uncertainty in the predictions, which can be useful in decision making.

What are some common techniques for parameterizing conditional expectations in the Gaussian case?

Some common techniques include linear regression, logistic regression, and probit regression. These methods involve fitting a linear or nonlinear model to the data and using the parameters of the model to make predictions about the conditional expectations.

How does parameterizing conditional expectations in the Gaussian case differ from other methods?

Parameterizing conditional expectations in the Gaussian case differs from other methods in that it specifically uses the Gaussian distribution to model the relationship between variables. This allows for more flexibility and accuracy in the predictions compared to other methods that may assume a different type of distribution.

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