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- Question : 1SQ - What is wrong with the following claims? (a)
- Question : 2SQ - A baseball batter Tim has a better batting average than his teammate Frank. However, someone notices that Frank has a better batting average than Tim against both right-handed and left-handed pitchers. How can this happen? (Present your answer in a table.)
- Question : 3SQ - Determine, for each of the following causal stories, whether you should use the aggregate or the segregated data to determine the true effect. (a) There are two treatments used on kidney stones: Treatment A and Treatment B. Doctors are more likely to use Treatment A on large (and therefore, more severe) stones and more likely to use Treatment B on small stones. Should a patient who doesn
- Question : 4SQ - In an attempt to estimate the effectiveness of a new drug, a randomized experiment is conducted. In all, 50% of the patients are assigned to receive the new drug and 50% to receive a placebo. A day before the actual experiment, a nurse hands out lollipops to some patients who show signs of depression, mostly among those who have been assigned to treatment the next day (i.e., the nurse
- Question : 1SQ - Identify the variables and events invoked in the lollipop story of Study question 1.2.4
- Question : 1SQ - Consider Table 1.5 showing the relationship between gender and education level in the U.S. adult population. (a) Estimate P(High School). (b) Estimate P(High School OR Female). (c) Estimate P(High School | Female). (d) Estimate P(Female | High School). Table 1.5 The proportion of males and females achieving a given education level Gender Highest education achieved Occurrence (in hundreds of thousands) Male Never finished high school 112 Male High school 231 Male College 595 Male Graduate school 242 Female Never finished high school 136 Female High school 189 Female College 763 Female Graduate school 172
- Question : 1SQ - Consider the casino problem described in Section 1.3.6 (a) Compute P(
- Question : 1SQ - Suppose we have three cards. Card 1 has two black faces, one on each side; Card 2 has two white faces; and Card 3 has one white face and one back face. You select a card at random and place it on the table. You find that it is black on the face-up side. What is the probability that the face-down side of the card is also black? (a) Use your intuition to argue that the probability that the face-down side of the card is also black is 1 2 . Why might it be greater than 1 2 ? (b) Express the probabilities and conditional probabilities that you find easy to estimate (for example, P(CD = Black)), in terms of the following variables: I = Identity of the card selected (Card 1, Card 2, or Card 3) CD = Color of the face-down side (Black, White) CU = Color of the face-up side (Black, White) Find the probability that the face-down side of the selected card is black, using your estimates above. (c) Use Bayes
- Question : 1SQ - Prove, using Bayes
- Question : 1SQ - (a) Prove that, in general, both ??XY and ??XY vanish when X and Y are independent. [Hint: Use Eqs. (1.16) and (1.17).] (b) Give an example of two variables that are highly dependent and, yet, their correlation coefficient vanishes.
- Question : 1SQ - Two fair coins are flipped simultaneously to determine the payoffs of two players in the town
- Question : 1SQ - Compute the following theoretical measures of the outcome of a single game of craps (one roll of two independent dice), where X stands for the outcome of Die 1, Z for the outcome of Die 2, and Y for their sum. (a) E[X], E[Y], E[Y|X = x], E[X|Y = y], for each value of x and y, and Var(X), Var(Y), Cov(X, Y), ??XY ,Cov(X, Z) Table 1.6 describes the outcomes of 12 craps games. (b) Find the sample estimates of the measures computed in (a), based on the data from Table 1.6. [Hint: Many software packages are available for doing this computation for you.] (c) Use the results in (a) to determine the best estimate of the sum, Y, given that we measured X = 3. Table 1.6 Results of 12 rolls of two fair dice X ZY Die 1 Die 2 Sum Roll 1 6 3 9 Roll 2 3 4 7 Roll 3 4 6 10 Roll 4 6 2 8 Roll 5 6 4 10 Roll 6 5 3 8 Roll 7 1 5 6 Roll 8 3 5 8 Roll 9 6 5 11 Roll 10 3 5 8 Roll 11 5 3 8 Roll 12 4 5 9 (d) What is the best estimate of X, given that we measured Y = 4? (e) What is the best estimate of X, given that we measured Y = 4 and Z = 1? Explain why it is not the same as in (d).
- Question : 1SQ - (a) Prove Eq. (1.22) using the orthogonality principle. [Hint: Follow the treatment of Eq. (1.26).] (b) Find all partial regression coefficients RYX?Z, RXY?Z, RYZ?X, RZY?X, RXZ?Y , and RZX?Y for the craps game described in Study question 1.3.7. [Hint: Apply Eq. (1.27) and use the variances and covariances computed for part (a) of this question.]
- Question : 1SQ - Consider the graph shown in Figure 1.8: XY Z T W Figure 1.8 A directed graph used in Study question 1.4.1 (a) Name all of the parents of Z. (b) Name all the ancestors of Z. (c) Name all the children of W. (d) Name all the descendants of W. (e) Draw all (simple) paths between X and T (i.e., no node should appear more than once). (f) Draw all the directed paths between X and T.
- Question : 1SQ - Suppose we have the following SCM. Assume all exogenous variables are independent and that the expected value of each is 0. SCM 1.5.3 V = {X, Y, Z}, U = {UX, UY, UZ}, F = {fX, fY , fZ} fX ? X = uX fY ? Y = X 3 + UY fZ ? Z = Y 16 + UZ (a) Draw the graph that complies with the model. (b) Determine the best guess of the value (expected value) of Z, given that we observe Y = 3. (c) Determine the best guess of the value of Z, given that we observe X = 3. (d) Determine the best guess of the value of Z, given that we observe X = 1 and Y = 3. (e) Assume that all exogenous variables are normally distributed with zero means and unit variance, that is, ?? = 1. (i) Determine the best guess of X, given that we observed Y = 2. (ii) (Advanced) Determine the best guess of Y, given that we observed X = 1 and Z = 3. [Hint: You may wish to use the technique of multiple regression, together with the fact that, for every three normally distributed variables, say X, Y, and Z, we have E[Y|X = x, Z = z] = RYX?Zx + RYZ?Xz.] (f) Determine the best guess of the value of Z, given that we know X = 3.
- Question : 1SQ - Assume that a population of patients contains a fraction r of individuals who suffer from a certain fatal syndrome Z, which simultaneously makes it uncomfortable for them to take a life-prolonging drug X (Figure 1.10). Let Z = z1 and Z = z0 represent, respectively, the presence and absence of the syndrome, Y = y1 and Y = y0 represent death and survival, respectively, and X = x1 and X = x0 represent taking and not taking the drug. Assume that patients not carrying the syndrome, Z = z0, die with probability p2 if they take the drug and with probability p1 if they don
- Question : 1SQ - Consider a graph X1 ? X2 ? X3 ? X4 of binary random variables, and assume that the conditional probabilities between any two consecutive variables are given by P(Xi = 1|Xi?1 = 1) = p P(Xi = 1|Xi?1 = 0) = q P(X1 = 1) = p0 Compute the following probabilities P(X1 = 1, X2 = 0, X3 = 1, X4 = 0) P(X4 = 1|X1 = 1) P(X1 = 1|X4 = 1) P(X3 = 1|X1 = 0, X4 = 1)
- Question : 1SQ - Define the structural model that corresponds to the Monty Hall problem, and use it to describe the joint distribution of all variables.

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