Problem Set 1

# Problem Set 1 - Problem Set 1 Solutions Econ 120C Winter...

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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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Problem Set 1 - Problem Set 1 Solutions Econ 120C Winter...

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