need help with this question. thank you

This question will discuss the consequences of nonrepresentative sampling (recall the Dewey versus Truman example , and how we can correct for it in certain cases by reweighting the sample pOpulation. Suppose each individual in the population of interest is characterized by an outcome of interest Y, (e. g. did you vote for Truman), some observable characteristics X, (eg. income), and an indicator for whether they are part of the surveyed population 3,: (e. g. do you have a phone in 1948). We are interested in Elli], but even if we took an innitely large survey, we would only directly measure E [Yglsi = 1], since people with 3,; = D are never surveyed. (d) Suppose again that 31.: J. Y,|X,_. Show that Elms: = 1] = / Elmlxi = mlmls = 1m, where f(m|S,; = 1) is the conditional density of Xngg; = l. [Hint: use the law of iterated expecta tions, the conditional independence assumption, and the denition of the conditional expectation]. (e) The expression you derived in the previous question implies that if we consider the reweighted outcome, 1",; = w(X,;)Y,;, then Elms = 11: [mass = mmmls = mm. For what choice of 10(3) does E[17];|Si = 1] = E[Y,;]? (f) For which observations are the weights w(X,;) derived in the previous question large/Small? In the Dewey versus Truman example, how would the weights relate to income? (g) What condition do you need so that the denominator in the weights that you derived above is nonzero? Intuitively, what does this mean in the Dewey versus Truman example