3. [Hierarchical models] Some economic model assumes that individuals are heterogeneous in their productivity factor. This factor can range between 0 and 1, and is uniformly distributed across the economy. Thus, the productivity factor of a randomly selected individual is modeled with the uniform random variable, X ~ U[0, 1]. An output produced by an individual is random, but depends on the productivity, Y|X ~ B(n, X), where n is some natural number. 1 (a) Using the Law of Total Expectations find E(Y). (b) Using the Law of Total Variances, find V(Y).

Olympic Pole Vault The graph in Figure 7 indicates that in recent years the winning Olympic men’s pole vault height has fallen below the value predicted by the regression line in Example 2. This might have occurred because when the pole vault was a new event there was much room for improvement in vaulters’ performances, whereas now even the best training can produce only incremental advances. Let’s see whether concentrating on more recent results gives a better predictor of future records. (a) Use the data in Table 2 (page 176) to complete the table of winning pole vault heights shown in the margin. (Note that we are using x=0 to correspond to the year 1972, where this restricted data set begins.) (b) Find the regression line for the data in part ‚(a). (c) Plot the data and the regression line on the same axes. Does the regression line seem to provide a good model for the data? (d) What does the regression line predict as the winning pole vault height for the 2012 Olympics? Compare this predicted value to the actual 2012 winning height of 5.97 m, as described on page 177. Has this new regression line provided a better prediction than the line in Example 2?

Suppose employers care about overall productivity Z, which is made up of two components, X and Y, such that Z=X+Y (and employers know this productivity function). Suppose there are two groups (male and female) whose average Zs are the same (i.e. Z_M =Z_F), but X_M < X_F and Y_M > Y_F (and again, employers know these population averages). (Averages are given by bold font.) Researchers decide to run a resume-based audit study where they randomize names based on gender (using typically male or female names) but hold constant Y. They provide no information on X. Which of the following results would provide suggestive evidence of statistical discrimination?  A. The call-back rate is the same for both sexes. B. We can't use differences in call-back rates to find evidence of statistical discrimination.C. The call-back rate is higher for women than for men. D. The call-back rate is higher for men than for women.

Consider house rent prediction problem where you are supposed to predict price of a house based on just its area. Suppose you have n samples with their respective areas, x(¹), x(²),...,x(n), their true house rents y(¹), y(²),..., y(n). Let's say, you train a linear regressor that predicts ƒ (x(i)) : = 0 + 0₁x¹). The parameters and ₁ are scalars and are learned by minimiz- ing mean-squared-error loss through gradient descent with a learning rate a. Answer the following questions. 1. Express the loss function(L) in terms of x(i),y(i), n, 00, 0₁. 2. Compute L მძი ƏL 3. Compute 20₁ 4. Write update rules for 0 and 0₁