{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

05. Moment generating functions and properties

# 05. Moment generating functions and properties - University...

This preview shows pages 1–3. Sign up to view the full content.

University of California, Los Angeles Department of Statistics Statistics 100B Instructor: Nicolas Christou Moment generating functions Definition: M X ( t ) = Ee tX Therefore, If X is discrete M X ( t ) = X x e tX P ( x ) If X is continuous M X ( t ) = Z x e tX f ( x ) dx Aside: e x = 1 + x 1! + x 2 2! + x 3 3! + · · · Similarly, e tx = 1 + tx 1! + ( tx ) 2 2! + ( tx ) 3 3! + · · · Let X be a discrete random variable. M X ( t ) = X x e tX P ( x ) = X x " 1 + tx 1! + ( tx ) 2 2! + ( tx ) 3 3! + · · · # P ( x ) or M X ( t ) = X x P ( x ) + t 1! X x xP ( x ) + t 2 2! X x x 2 P ( x ) + t 3 3! X x x 3 P ( x ) + · · · To find the k th moment simply evaluate the k th derivative of the M X ( t ) at t = 0. EX k = [ M X ( t )] k th derivative t =0 For example: First moment: M X ( t ) 0 = X x xP ( x ) + 2 t 2! X x x 2 P ( x ) + · · · We see that M X (0) 0 = x xP ( x ) = E ( X ). 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Similarly, Second moment M X ( t ) 00 = X x x 2 P ( x ) + 6 t 3! X x x 3 P ( x ) + · · · We see that M X (0) 00 = x x 2 P ( x ) = E ( X 2 ). Examples: Find the moment generating function of X b ( n, p ). Find the moment generating function of X Poisson ( λ ).
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern