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Unformatted text preview: Problem Set 1 Solutions Econ 120C / Winter 2008 Due date: January 23, 2008 1 Exercises The exercises in this problem set are similar to those in Chapter 2 of Stock and Watson. You may benefit by referring to exercises 2.1, 2.12, 2.21 and 2.23 to get an idea of how to approach the problem set exercises. 1. Let Y denote a random variable for the number of “heads” that occur when two coins are tossed. One of the coins is fair, and the second is not; for the second coin, the probability of heads is 2/5. Assume the outcomes of the two coin tosses are independent. (a) Derive the probability distribution of Y . We derive the probability distribution of Y by looking at all possible ways to get each out come. Because the coin tosses are independent, the probabilities follow the multiplication rule for independent events; for instance, Pr ( H,T ) = Pr ( H ) Pr ( T ) . We can summarize the information as follows: Fair Unfair # Heads Probability H H 2 1 / 2 · 2 / 5 = 1 / 5 H T 1 1 / 2 · 3 / 5 = 3 / 10 T H 1 1 / 2 · 2 / 5 = 1 / 5 T T 1 / 2 · 3 / 5 = 3 / 10 This means that the probability distribution is as follows: f ( y ) = 3 / 10 if y = 0 1 / 2 if y = 1 1 / 5 if y = 2 1 (b) Derive the cumulative probability distribution of Y . The cumula tive probability distribution F ( y ) , is defined as F ( y ) = Pr ( Y ≤ y ) . We can derive this from the probability distribution, f , in part (a) above. So, the cumulative probability distribution of Y is: F ( y ) = if y < 3 / 10 if 0 ≤ y < 1 4 / 5 if 1 ≤ y < 2 1 if 2 ≤ y (c) Derive the mean and variance of Y . These are straightforward calculations. The variance calculation uses a formula proved in part 3a. We have that: E ( Y ) = 0 · 3 / 10 + 1 · 1 / 2 + 2 · 2 / 10 = 5 / 10 + 4 / 10 = 9 / 10, and var ( Y ) = E ( Y 2 ) E ( Y ) 2 = (0 · 3 / 10 + 1 · 1 / 2 + 4 · 2 / 10) (9 / 10) 2 = 5 / 10 + 8 / 10 81 / 100 = 130 / 100 81 / 100 = 49 / 100 2. Compute the following probabilities. We can compute these probabil ities by interpolating statistical tables such as those found in the Ap pendix in Stock and Watson, or by using a statistical software package such as Stata. Here, we use interpolation for (a), tables for (b), and statistical software for (c) and (d). (a) If Y is distributed t 10 , find Pr( Y > 1 . 75). From the tables in the Appendix of Stock and Watson, we know that the 10% and 5% critical values of a t 10 distribution are 1.37 and 1.81 respectively. Interpolation is simply a linear approximation in this region. We wish to solve the following equation: α · 1 . 37+(1 α ) · 1 . 81 = 1 . 75 . This is solved at α = 3 22 . So, Pr( Y > 1 . 75) = 19 22 · . 05 + 3 22 · . 1 ≈ . 0568 ....
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 Winter '08
 Stohs
 Econometrics, Variance, probability density function

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