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Unformatted text preview: Econ 103 UCLA, Fall 2010 Answers to the Problem Set 1 by Dmitry Plotnikov Part 1: True or False and explain briefly why. 1. The expected value of a discrete random variable is the outcome that is most likely to occur. FALSE. The expected value of a random variable can lie (and usually does) between the possible outcomes of possible outcomes of a discrete random variable. This happens because by definition it is a weighted average of these outcomes. 2. If two random variables X and Y are independently distributed, then E ( Y ) = E ( Y  X ) . TRUE. If X and Y are independently distributed, distribution of Y does not depend on X , thus E ( Y  X ) can not depend on X and has to be equal to E ( Y ) 3. A probability density function tells the probability that a random variable is less than or equal to a certain value. FALSE. It is a cumulative distribution function that tells the probability that a random variable is less than or equal to a certain value. 4. V ar ( X + Y ) = V ar ( X ) + V ar ( Y ) + 2 Cov ( X,Y ) TRUE. Follows from the definition of variance. 5. V ar ( X Y ) = V ar ( X ) V ar ( Y ) 2 Cov ( X,Y ) FALSE. V ar ( X Y ) = V ar ( X )+ V ar ( Y ) 2 Cov ( X,Y ) because V ar ( Y ) = V ar ( Y ) . 6. If XY = 0 , then X and Y are independent. FALSE. If two random variables are uncorrelated it does not mean they are independent. However the opposite is true. 7. Let Y be a random variable. Then the standard deviation of Y equals E ( Y Y ) . FALSE. Using properties of expectation operator E ( Y Y ) = E ( Y ) Y = Y Y = 0 .Also, by definition, the standard deviation of Y is equal to Y = p E [( Y Y ) 2 ] . 8. Assume that X, Y and Z follow the distribution N ( ; 2 ) . Then W = X + Y Z is normally distributed. TRUE. Any linear combination of normal random variables is normally distributed. 9. Assume that Y F 1 , . Then Y 2 1 . TRUE. Property of Fdistribution. Replacing m = 1 in F m, = 2 m (see Lecture 1) the result follows. 1 Econ 103 UCLA, Fall 2010 10. Observations in a random sample are independent of each other. TRUE. Definition of a random sample. 11. If is an unbiased estimator of , then = . FALSE. If is an unbiased estimator of , then E [ ] = , but in general will not exactly equal . 12. If the pvalue equals 0.96, then we cannot reject the null hypothesis. TRUE. We cannot reject the null if pvalue is greater than the significance level (which usually equals 0.01, 0.05 or 0.10) 13. The standard error of Y equals the standard deviation of Y . That is, SE ( Y ) = Y . FALSE. It was calculated in class that V ar ( Y ) = 2 Y n . Thus SE ( Y ) = Y n 14. Assume that H : Y = Y, and H 1 : Y > Y, , and Y is normally distributed. To compute the critical value for this 1sided test, we divide by two the positive critical value of the 2sided test....
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 Winter '09
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