• Share
  • Email
  • Embed
  • Like
  • Save
  • Private Content
Moment Generating Functions
 

Moment Generating Functions

on

  • 1,914 views

Moment Generating Functions

Moment Generating Functions

Statistics

Views

Total Views
1,914
Views on SlideShare
1,900
Embed Views
14

Actions

Likes
0
Downloads
0
Comments
0

3 Embeds 14

http://www.slideshare.net 7
http://dataminingtools.net 4
http://www.dataminingtools.net 3

Accessibility

Categories

Upload Details

Uploaded via as Microsoft PowerPoint

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

    Moment Generating Functions Moment Generating Functions Presentation Transcript

    • 1.6 Moment Generating Functions
    • Moment Generating Functions (mgf)
      This is called the kth raw moment or kth moment
      about the origin.
      is the first moment about origin
      This implies that
      or first raw moment.
    • Is called the kth moment about the mean
      or the kth central moment.
      Therefore
      is called the second central moment.
      Moment Generating Functions
      • The mean  is the first moment about origin and variance is the second moment about the mean.
      • Higher moments are often used in statistics to give further descriptions of the probability distributions.
      Moment Generating Functions
    • Moment Generating Functions
      The third moment about the mean is used to describe the symmetry or skewness of a distribution.
      The fourth moment about mean is used to describe its “peakedness” or kurtosis.
      Kurtosis is a quantity indicative of the general form of a statistical frequency curve near the mean of the distribution.
    • Given below is the density of X
      Find
      Moment Generating Functions
    • Moment Generating Functions
    • Moment Generating Functions
    • Definition: Moment Generating Function
      Let X be a random variable with density f. The moment generating function of X (mgf) is denoted by
      and is given by
      Provided this expectation is finite for all real numbers
      in some open interval
    • Theorem: Let
      be the moment generating function
      for a random variable
      Then,
      Moment Generating Functions
    • Moment Generating Functions
      The moment generating function is unique and completely determines the distribution of the random variable; thus if two random variables have the same mgf, they have the same distribution (density). Proof of uniqueness of the mgf is based on the theory of transforms in analysis, and therefore we merely assert this uniqueness.