I could not clearly understand the aim of the question as shown in the image. I will list the steps on how to deal with this problem. If there is any mistake and errors in them, please make a correction.

First, because every row has independent variables (?x1??, ?x2?? ), which generate 200 (Z values), I will put the values to the formula introduced in Y.

Second, ?Yi?? for each row is a numerical value rather than a vector, and I will use proper regression models to capture a relationship between X(ie ?x1??, ?x2?? ) and Y.

Third, using any computation statistical packages, I will obtain the mean and variance for Y.

That is all I can do about this problem. Hope to hear your advice on this problem.

Overview: In probability and statistics, it is important to understand the mean and variance for any random variables. In this exam, suppose that Y = V(X1, X2) is a random variable whose distribution depends on two independent variables X, and X2, and the objective is to estimate two deterministic functions of X] and X2: one is the mean a(X1, X2) = E(Y) and the other is the variance V(X1, X2) = Var(Y). Background: The motivation of the random variable Y = Y(X1, X2) is from overshoot analysis of random walks in applied probability and renewal theory. For a given pair of (X1, X2), one first generates a m-dimensional vector, (21, Z2, . .., Zm), say, m = 50, which has a multivariate Gaussian distribution with m-dimensional mean vector and m x m covariance matrix depending on both X] and X2. Then the random variable Y = Y(X1, X2) is defined as Y = Y(X1, X2) = 100 * - max(e21, ezz. ..., ezm ) literaturtem er (1) Here we multiply by 100 since the numerical values of Y might be too small in our context otherwise. Clearly 0