econ6190 s6

econ6190 s6 - distribution does not have all of its...

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ECON 6190 TA Section 6 Jijie Zhao 10/7/2011 1 Moment generating functions Moment generating function (mgf): The mgf of a rv X M X ( t ) = E [exp( tX )] = x exp( tx ) f X ( x ) drv R 1 exp( tx ) f X ( x ) dx crv: : If the expectation exists for t in some neighborhood of 0, then we say that M X ( t ) exists for t in a small neighborhood of 0. If the expectation does not exist for any small neighborhood of 0, then M X ( t ) does not exist for the distribution of X . MGF uniquely determines the distribution of a random variable: Suppose two r.v. X and Y with M X ( t ) and M Y ( t ) existing in a neighborhood of 0. Then X and Y have the same M X ( t ) and M Y ( t ) for all t in N (0) ; if and only if F X ( u ) = F Y ( u ) for all u 2 R : The k-th moment is equal to the kth derivative of the mgf evaluated at 0: M ( k ) X (0) = E ( X k ) : In particular, M X (0) = 1; M 0 X (0) = X ; M 00 X (0) = E ( X 2 ) : ( ± 2 = M 00 X (0) ± [ M 0 X (0)] 2 ) when a mgf exists, all the moments of the distribution can be calculated from this mgf; if the
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Unformatted text preview: distribution does not have all of its moments, it certainly does not have a mgf. 1.1 Examples 1. If a d.r.v. X has M X ( t ) = ( pe t + 1 ± p ) n ; what is the pmf? 2. Does a distribution exist for which M X ( t ) = t= (1 ± t ) ; j t j < 1 ? If yes, &nd it. If no, prove it. 3. In each of the following cases verify the expression given for the moment generating function, and in each case use the mgf to calculate EX and V arX . (a) P ( X = x ) = e & & ± x x ! , M X ( t ) = e ± ( e t & 1) ; x = 0 ; 1 ; 2 ;::: ; ² > 0; (b) f X ( x ) = e & ( x & ± ) 2 = (2 ² 2 ) p 2 ²³ , M X ( t ) = e ´t + ³ 2 t 2 = 2 , ±1 < x < 1 ; ±1 < & < 1 ; ± > . 1...
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This note was uploaded on 01/07/2012 for the course ECON 6190 taught by Professor Hong during the Fall '07 term at Cornell.

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