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Econ375Homework1Solution

Econ375Homework1Solution - ECON 375 Introduction to...

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ECON 375: Introduction to Econometrics Spring 2010 Problem Set 1 Due: April 12, 2010 NOTE: All assignments must be typed if you can not write it legibly. Graphs and calculations may be handwritten. No late homework will be accepted. Exercise 1 (10 Points) A fair six-sided dice is thrown and the result is recorded. The same die is thrown a second time. Calculate the probability that the num- ber obtained on the second toss exceeds the number obtained on the first toss. Solution: Sample space consists of 36 = 6 6 elements which are tuples of toss results such as ( x, y ) = (1 , 1), ( x, y ) = (2 , 2) where x represents the result of the first toss, y represents the result of the second toss. We would like to have y > x . We can count the number of such occurrences by first assuming a result for the first toss, x . If x = 1, then y = 2 , 3 , 4 , 5 , 6 would satisfy our condition, there are 5 such cases for x = 1. Similarly for x = 2 there are 4 such cases, for x = 3 there are 3, for x = 4 there are 2, for x = 5 there is 1 and for x = 6 there are none. This means we have total 5 + 4 + 3 + 2 + 1 = 15 cases. This means probability is 15 / 36. Exercise 2 (10 Points) Bart is planning to murder his rich Uncle Basil in hopes of claiming his inheritance a bit early. Hoping to take advantage of his uncles predilection for immoderate deserts, Bart has put rat poison in the cherries flambe and cyanide in the chocolate mousse. The probability of the rat poison being fatal is 60 percent; the cyanide 90 percent. Bart estimates that Basil has a 60 percent chance of ordering the cherries and a 40 percent chance of ordering the mousse. Given that Basil did, indeed, suffer a premature demise, what is the probability that he was done in by the chocolate mousse? Solution: We are given the following probabilities: P ( D | C ) = 0 . 6, P ( D | M ) = 0 . 9, P ( C ) = 0 . 6 and P ( M ) = 0 . 4. Here D denotes death, C denotes eating cherries, M denotes eating mousse. We want to find P ( M | D ). Using Bayesian formula we get 1
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P ( M | D ) = P ( M D ) P ( D ) We can find P ( M D ) = P ( D | M ) P ( M ) = 0 . 9 0 . 4 = 0 . 36. We need to find P ( D ). Uncle eats either cherries or flambe, so his chance of death will be P ( D ) = P ( C ) P ( D | C ) + P ( M ) P ( D | M ) = 0 . 6 0 . 6 + 0 . 4 0 . 9 = 0 . 72 So, answer is P ( M | D ) = 0 . 36 / 0 . 72 = 0 . 5 Exercise 3 (10 Points) The rated capacity for an elevator at the Sears Tower is 2500 kilograms. If this capacity is exceeded the elevator cable will snap and will plummet to the ground. Varsity football players have weights that are normally distributed with a mean of 120 kilograms and a standard deviation of 80 kilograms. Saturday night the team goes disco dancing at the Sears Tower. After closing, 25 drunken football players get on the elevator. What is the likelihood that the elevator cable will snap? Solution: One of the most important things you need to know and remem- ber about this course is that sum of normal random variables gives you another normal random variable, the other one is square of a normal random variable is Chi-Square.
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