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# In probability and statistics, it is important to understand the mean and variance for any random variable

INSTRUCTIONS TO CANDIDATES

This is an open-book take-home final exam. You are free to use any recourses including textbooks, notes, computers and internet, but no collaborations are allowed, particularly you cannot commu- nicate, online or orally, with any other people about this midterm (except the TAs or instructor via piazza if you have any questions or concerns). This must be individual work. See Canvas for the data sets and some useful R codes.

Overview: In probability and statistics, it is important to understand the mean and variance for any random variables. In many applications, it is straightforward to simulate the random variable Y ’s, but it is often highly non-trivial to characterize the exact distribution  of  Y =  Y (X1, X2)  including deriving the explicit formulas for the mean and variance of Y = Y (X1, X2) explicitly as a function of X1 and X2.

Objective: In this exam, suppose that Y = Y (X1, X2) is a random variable whose distribution depends on two independent variables X1 and X2, and the objective is to estimate two deterministic functions of X1  and X2:  one is the mean µ(X1, X2) = E(Y ) and the other is the variance V (X1, X2) = V ar(Y ).

For that purpose, you are provided the observed 200 realizations of the Y ’s values for some given pairs (X1, X2)’s. You are asked to use data mining or machine learning methods that allow us to conveniently predict or approximate the mean  and  variance  of  Y =  Y (X1, X2)  as  a  function  of  X1 and  X2.  That  is,  your  task  is  to  predict  or  approximate  two  values  for  those  given  pairs  (X1, X2) in the  testing  data  set:  one  for  the  mean  µ(X1, X2)  =  E(Y (X1, X2))  and  the  other  for  the  variance V (X1, X2) = V ar(Y (X1, X2)).

Training data set: In order to help you to develop a reasonable estimation of the mean and variance  of  Y    =   Y (X1, X2)  as  deterministic  functions  of  X1   and  X2,  we  provide  a  training  data set that is generated as follows.  We first choose the uniform design points when 0 ≤ X1 ≤ 1 and 0 ≤ X2 ≤ 1, that is, x1i = 0.01 ∗ i for i = 0, 1, 2, . . . , 99, and x2j = 0.01 ∗ j for j = 0, 1, 2, . . . , 99. Thus there are a total of 100 ∗ 100 = 104 combinations of (x1i, x2j)’s, and for each of these 104 combinations, we generate 200 independent realizations of the Y variables, denoted by Yijk for k = 1, . . . , 200.

The corresponding training data, 2022Fall7406train.csv,  is available from Canvas.  Note that this training data set is a 104 × 202 table. Each row corresponds to one of 100 ∗ 100 = 104 combinations of (X1, X2)’s. The first and second columns are the X1 and X2 values,  respectively,  whereas  the remaining 200 columns are the corresponding 200 independent realizations of Y ’s.

Based on the training data, you are asked to develop an accurate estimation of the functions µ(X1, X2) = E(Y ) and V (X1, X2) = V ar(Y ), as deterministic functions of X1 and X2 when 0 ≤ X1 ≤ 1 and 0 ≤ X2 ≤ 1.

To assist you, a limited empirical data analysis (EDA) on the training data is provided in the

appendix by using R. Please feel free to modify to other language such as Python, Matlab, etc.

Testing data set: For the purpose of evaluating your proposed estimation models and methods, we choose 50 random design points for X1 and 50 random design points for X2. Thus there are a total of 50 ∗ 50 = 2500 combinations of (X1, X2) in the testing data set. You are asked to use your formula to predict µ(X1, X2) = E(Y ) and V (X1, X2) = V ar(Y ) for Y = Y (X1, X2) for the 50 ∗ 50 = 2500 combination of (X1, X2) in the testing data (please keep the six digits for your answers).

The exact values of the (X1, X2)’s in the testing data set are included in the file 2022Fall7406test.csv, which is available from Canvas. You are asked to use your formula to predict µ(X1, X2) = E(Y ) and

V (X1, X2) = V ar(Y ) for the 50 ∗ 50 = 2500 combination of  (X1, X2)  in  the  testing  data  (please  keep (at least) six digits for your answers).

Estimation Evaluation Criterion: In order to evaluate your estimation or prediction, we obtain “true” values µ(X1, X2) = E(Y ) and V (X1, X2) = V ar(Y ) for each combination of (X1, X2) in the testing data set, based on the following Monte Carlo simulations (we will not release these true values!). We first generated 200 random realizations of Y ’s for each combination of (X1, X2) in the testing data set, but we will not release these 200 independent realizations for the testing data. Next, for each given combination of (X1, X2), we have 200 realizations of Y ’s, denoted by Y1, • • • , Y200, and then we

compute the “true” values as

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