# Question:How to optimize an expression with conditional means and variances?

## Question:How to optimize an expression with conditional means and variances?

Maple 2022

Hello everyone,

Here is a stylized version of my problem. Given three normally distributed random variables {A, B, C}, I want to find the X_1, X_2, X_3 that maximize the following expression (gamma being a constant and A being a linear combination of other normally distributed random variables):

Max{ Exp[A|B,C] - (gamma/2)*Var[A|B,C] }

All the details are here: 070423_Optimization.mw

In particular, I am seeking your help to:

1. Correlate three random variables (an example of the procedure for correlating two random variables is already provided in the script).
2. Verify my understanding of the linear projection theorem (https://stats.stackexchange.com/questions/30588/deriving-the-conditional-distributions-of-a-multivariate-normal-distribution) in two dimensions, that is, to compute conditional means and variances of the form E[X|Y,Z] and V[X|Y,Z].
3. Implement and adapt the linear projection theorem to my problem in Maple.
4. Combine all together to obtain Expr = E[A|B,C] + V[A|B,C] and find the optimal {X_1, X_2, X_3} by solving the linear system of three equations in the three variables {X_1, X_2, X_3}, where the three equations are obtained by setting to 0 the partial derivatives of Expr with respect to {X_1, X_2, X_3}.

In relation to point 2., did I correctly interpret the matrix form of the three-dimensions version of the linear projection theorem?

In relation to point 3., I attach a stylized script for the three-dimensions version (note that I need the two-dimensions version for my problem): LinearProjectionTheorem_3dimensions.mw. Assuming a correct interpretation of the theorem:

• Did I correctly implement E[X_2|Y_1,Y_2,Y_3] and E[X_3|Y_1,Y_2,Y_3] as in the picture above?
• How to adapt it accordingly to include E[X_1|Y_1,Y_2,Y_3] and V[X_1|Y_1,Y_2,Y_3], V[X_2|Y_1,Y_2,Y_3], and V[X_3|Y_1,Y_2,Y_3]?
• How to apply it to the random variables in my script to eventually find the optimal {X_1,X_2,X_3}?

You can play around with my script 070423_Optimization.mw and send me the updated version. The problem I am trying to solve is quite convoluted, so let me know if you need any further clarification. Thanks a lot!

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