econ483prob01stat - X ³ N&± 2 ± Suppose that we have...

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Econ 483 °Problem Set 1 1. Suppose that the probability density function of a random variable X is given by p ( x ) = j x j ° 1 [ ± 1 ² x ² 1] . In other words, p ( x ) = ± x if ± 1 ² x ² 0 , p ( x ) = x if 0 ² x ² 1 , and p ( x ) = 0 otherwise. Let ° = E ( X ) and ± 2 = V ar ( X ) . (1) Show that the expectation of X is 0 . In other words, show that ° = 0 . (2) Show that the variance of X is 1 2 . In other words, show that ± 2 = 1 2 . Suppose that we have 2 random draws, X 1 and X 2 . Let X 2 = 1 2 ( X 1 + X 2 ) . (3) What is the expectation of X 2 ? (4) Explain why the covariance of X 1 and X 2 is 0 . (5) What is the variance of X 2 ? (6) Is X 2 a random variable? How about E ° X 2 ± and V ar ° X 2 ± ? Suppose that we have n random draws, X 1 , X 2 , ..., X n . Let X = 1 n P n i =1 X i . (7) What is the expectation of X ? (8) What is the variance of X ? (9) As n approaches in°nity, V ar ° X ± converges to 0 . How is this fact related to the law of large numbers? 2. Suppose that the probability density function (or the likelihood function) of a random variable X is given by p ( x ) = p ° x; ± 2 ± = 1 p 2 °± 2 e ° x 2 2 ° 2 . In other words, the random variable X follows a normal distribution with expectation 0 and variance ± 2 . So,
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Unformatted text preview: X ³ N & ;± 2 ± . Suppose that we have n random draws, X 1 , X 2 , ..., X n . (1) Since E ( X ) = 0 , we have ± 2 = E & X 2 ± . Obtain a sample analogue estimator of ± 2 . (2) Show that the maximum likelihood estimator of ± 2 is 1 n P n i =1 X 2 i . (3) Is 1 n P n i =1 X 2 i an unbiased estimator of ± 2 ? (4) Is 1 n P n i =1 X 2 i a consistent estimator of ± 2 ? 3. Suppose that the probability density function of a random variable X is given by p ( x ) = p ( x;² ) = ² x (1 ± ² ) 1 & x if x = 0 or 1 , and p ( x ) = p ( x;² ) = 0 otherwise. Suppose that we have n random draws, X 1 , X 2 , ..., X n . (1) Show that the expectation of X is ² . (2) Show that the maximum likelihood estimator of ² is b ² = 1 n P n i =1 X i . (3) Is b ² an unbiased estimator of ² ? (4) Is b ² a consistent estimator of ² ? 1...
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