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#### 1 Random Variables  1

###### Statistics

1 Random Variables
1.1 Suppose that a random variable X has a discrete distribution with the following p.m.f.: f (x) = {cx for x = 1, ..., 5,
() otherwise.
Determine the value of the constant c. [4 points]
1.2
Suppose that the p.d.f. of a random variable X is as follows:
f (x) = Icx2 for 1 < x < 2, 0 otherwise.
1. Find the value of the constant c. You can leave the constant c expressed in terms of the integral. However, try to reduce it as much as possible. An useful property of integrals is: cf (x)dx = c ro. f (x)dx . [3 points] 2. Find the value of Pr(X > 3/2). Once again, you can leave the actual integral unsolved. (But of course you can always solve it if you want to, and I'll give you 1 extra point!) [3 points]
1.3
Suppose that a random variable X can take only the values -2, 0, 1, and 4, and that the probabilities of these values are as follows: Pr(X = -2) = 0.4, Pr(X = 0) = 0.1, Pr(X = 1) = 0.3, and Pr(X = 4) = 0.2. Sketch the c.d.f. of X. (You can do it in pen and paper and then scan the drawing, or in paint. But the command segments might help you if you want to sketch it in R.) [5 points]
2 Distributions
2.1
Suppose you had a Poisson process with intensity parameter A = 5. 1. What is the probability of getting exactly 7 events? [2 points] 2. What is the probability of getting exactly 3 events? [2 points] 3. These values are the same distance from the expected value of the Poisson distribution, so why are they different? [2 points]
2.2
Given the following PMF:
{ f (x) = ..)!(1)3 x = 0, 1, 2, 3 0 otherwise
1. Prove that this is in fact a PMF [HINT: You might have to check the Kolmogorov axioms] [4 points] 2. Find the expected value [3 points] 3. Find the variance; [3 points] 4. Derive the CDF [4 points]
3 Democratic Presidential Systems
In democratic presidential systems, the probability that a president resigns or is ousted before the end of his or her mandate is 0.25. In other words, the probability that a given president does not stay in office as long as the Constitution requires is 0.25. We study the 72 elected presidents that served over the last 20 years in all of the presidential democracies in the world.
1. Define the random variable for this study. What values can your random variable take? [4 points]
2. How do you believe the random variable you defined in (1) is distributed? Explain your choice.  Use mathematical notation to indicate the distribution of your random variable, including the value for the parameters. [4 points] 3. What is the probability of observing 10 presidents not finishing their term in office? (a) Calculate this probability using the p.d.f. (Include R code for algebraic calculations.) [3 points] (b) Calculate this probability using the appropriate R command. [3 points] 4. What is the probability of observing more than 60 presidents not finishing their term in office?
(a) Calculate this probability using the c.m.f. (Include R code for algebraic calculations.) [3 points] (b) Calculate this probability using the appropriate R command. [3 points]
4 R question
Solve only the numbers 1-4 of the Exercise 4.5.2 of Kosuke Imai's Quantitative Social Science: An Introduction book ("Election and conditional cash transfer program in Mexico"). In point 3) of this exercise, ignore the "If the results are different, which model fits the data better?" question. The dataset is available in Canvas (under Files, the name is progresa. csv). [25 points]
4.5.2 ELECTION AND CONDITIONAL CASH TRANSFER PROGRAM IN MEXICO In this exercise, we analyze the data from a study that seeks to estimate the electoral impact of Progresa, Mexico's conditional cash transfer program (CCT program)! The original study relied on a randomized evaluation of the CCT program in which eligible villages were randomly assigned to receive the program either 21 months (early Progresa) or 6 months (late Progresa) before the 2000 Mexican presidential election. The author of the original study hypothesized that the CCT program would mobilize voters, leading to an increase in turnout and support for the incumbent party (PRI, or Partido Revolucionario Institutional, in this case). The analysis was based on a sample of precincts that contain at most one participating village in the evaluation. The data we analyze are available as the CSV file progresa . csv. Table 4.11 presents the names and descriptions of variables in the data set. Each observation in the data represents a precinct, and for each precinct the file contains information about its treatment status, the outcomes of interest, socioeconomic indicators, and other precinct characteristics.
1. Estimate the impact of the CCT program on turnout and support for the incumbent party (PRI) by comparing the average electoral outcomes in the "treated" (early Progresa) precincts versus the ones observed in the "control" (late Progresa) precincts. Next, estimate these effects by regressing the out-come variable on the treatment variable. Interpret and compare the estimates under these approaches. Here, following the original analysis, use the turnout and support rates as shares of the eligible voting population ( t 2000 and pri200 Os, respectively). Do the results support the hypothesis? Provide a brief interpretation.
7 This exercise is based on the following articles: Ana de la 0 (2013) "Do conditional cash transfers affect voting behavior? Evidence from a randomized experiment in Mexico.. American Journal of Political Science, vol. 57, no. 1, pp. 1-14 and Kosuke 'mai, Gary King, and Carlos Velasco (2015) .Do nonpartisan programmatic policies have partisan electoral effects? Evidence from two large scale randomized experiments.. Working paper.
2. In the original analysis, the author fits a linear regression model that includes, as predictors, a set of pretreatment covariates as well as the treatment variable. Here, we fit a similar model for each outcome that includes the average poverty level in a precinct (a vgpove rt y), the total precinct population in 1994 (pobt ot 1994), the total number of voters who turned out in the previous election (vot os1994), and the total number of votes cast for each of the three main competing parties in the pre-vious election (pri1994 for PRI, pan1994 for Partido Accion Nacional or PAN, and prd1994 for Partido de la Revolucion Democratica or PRD). Use the same outcome variables as in the original analysis, which are based on the shares of the voting age population. According to this model, what are the estimated average effects of the pro-gram's availability on turnout and support for the incumbent party? Are these results different from those you obtained in the previous question?
3. Next, we consider an alternative, and more natural, model specification. We will use the original outcome variables as in the previous question. However, our model should include the previous election outcome variables measured as shares of the voting age population (as done for the outcome variables t1994, pr i1994s, pan1994s, and prdl 994 s) instead of those measured in counts. In addition, we apply the natural logarithmic transformation to the precinct population variable when including it as a predictor. As in the original model, our model includes the average poverty index as an additional predictor. Are the results based on these new model specifications different from those we obtained in the previous question? If the results are different, which model better fits the data?
4. We examine the balance of some pretreatment variables used in the previous analyses. Using box plots, compare the distributions of the precinct population (on the origi-nal scale), average poverty index, previous turnout rate (as a share of the voting age population), and previous PRI support rate (as a share of the voting age population) between the treatment and control groups. Comment on the patterns you observe.

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