{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Econometrics 10.7 notes 2

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

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 - ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern