319handout7

319handout7 - E X and E X 2(c Recall from handout 5...

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Econ 319, Fall 2008 TA: Simon Kwok Handout 7 1. Suppose that the joint pdf of X and Y is given by: f ( x; y ) = 8 xy 0 < y < x < 1 ; 0 otherwise : (a) Find the marginal density of X: (b) Find the conditional density of Y j X: (c) Find E ( Y j X ) : (d) Find V ar ( Y j X ) : 2. In handout 5, question 1, we considered the joint pdf of X and Y as follows: f ( x; y ) = 2 5 (2 x + 3 y ) 0 x; y 1 ; 0 otherwise : Find: (a) E ( X j Y ) ; (b) V ar ( X j Y ) : 3. (Generalization of Markov Inequality) Suppose that the support of a continuous random variable X is nonnegative (i.e. Pr( X ± 0) = 1 ) and that E ( X k ) exists for all k > 0 . Prove that, for all k > 0 and t > 0 : Pr( X ± t ) E ( X k ) t k : 4. The pdf of a random variable X which follows an exponential distribution with parameter ( & > 0 ) is given by: f ( x ) = x > 0 ; 0 otherwise. (a) Find the moment generating function (mgf) of X .
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Unformatted text preview: E ( X ) and E ( X 2 ) : (c) Recall from handout 5, question 2 that the lifetime X (in 1000 hours) of the light bulbs that Simon bought follows an exponential distribution with & = 1 2 . 1. What is the mean lifetime of a lightbulb? 2. What is the variance of the lifetime of a lightbulb? 5. Recall that the pdf of a random variable X which follows a standard normal distribution is given by, for ²1 < x < 1 : f ( x ) = 1 p 2 ± e & x 2 = 2 : (a) Find the mgf of X . (b) Find the &rst four moments of X . [In fact, the k th moment of X is E ( X k ) = & if k is odd, 1 ³ 3 ³ 5 ³ ´´´ ³ ( k ² 1) if k is even. ] 1...
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