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Econometrics 10.7 notes 2

# Econometrics 10.7 notes 2 - ECON 6190 TA Section 6 Jijie...

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Unformatted text preview: ECON 6190 TA Section 6 Jijie Zhao 10/7/2011 1 Moment generating functions because we’re? always iwlémw 7’ ih t z o Moment generating function (mgf): The mgf of a rv X is deﬁned as Mx(t) = E[exp(tX)] For 6 609/0 { ZZZ exp(ta:)fx (2:) dry [3°00 exp(t:r)f)g\$m)da: crv. ‘ q ‘ l w‘si \IFAVI'“ mm"! Mats -'~7 MGF . .RMeF “ml; _——7 W , ml“ mwl~ all: It all hwmvvwl 4%;er amt ——-—§ MGF olaow o 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. 9 (May with “011527! 0 MGF uniquely determines the distribution of a random variable: Suppose two r.v. X and Y with M X (t) and My (t) existing in a neighborhood of 0. Then X and Y have the same M X (t) and My(t) for all t in NE(O), if and only ‘if Fx(u) = Fy(u) for all u E R. o The k—th moment is equal to the kth derivative of the mgf evaluated at 0: Mylo» : E(X"). In particular, MX(0) = 1; M§((O) = ,uX; M§'{(0) = E(X2).(c72 = M§’((O) — [M§{(0)]2 o A distribution with all moments exist and are ﬁnite may not have a mgf.(lognormal) 9 when a mgf exists, all the moments of the distribution can be calculated from this mgf; if the 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 MX (t) 2 (pet + 1 ~ p)", what is the pmf? 2. Does a distribution exist for which M X(t) = t/ (1 — t), |t| < 1? If yes, ﬁnd 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 Va’r‘X. (a) P(X = x) = JQL, MX(t) = eA<8‘-1),z = o,1,2,.. .; ,\ > 0; . _ _ 1_ 2 ‘,2 (b) fX(x) :— M, MX(t) =__e”tf°2‘2/2, —oo_<_ac '< oo; —oo < p < 00; a > 0. 21m - ...
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