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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 ρ 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 μ Y ) . FALSE. Using properties of expectation operator E ( Y μ Y ) = E ( Y ) μ Y = μ Y μ Y = 0 .Also, by definition, the standard deviation of Y is equal to σ Y = p E [( Y μ Y ) 2 ] . 8. Assume that X, Y and Z follow the distribution N ( μ ; σ 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 ∼ χ 2 1 . TRUE. Property of Fdistribution. Replacing m = 1 in F m, ∞ = χ 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 ˆ μ is an unbiased estimator of μ , then ˆ μ = μ . FALSE. If ˆ μ is an unbiased estimator of μ , then E [ˆ μ ] = μ , but in general ˆ μ will not exactly equal μ . 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 α (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 ) = σ Y . FALSE. It was calculated in class that V ar ( ¯ Y ) = σ 2 Y n . Thus SE ( ¯ Y ) = σ Y √ n 14. Assume that H : μ Y = μ Y, and H 1 : μ Y > μ 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
 Ramirez
 Normal Distribution, Standard Deviation, Variance, Probability theory, probability density function

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