Thread: [SOLVED] generating functions

can someone pls tell me the steps or ideas or materials or hints on how to obtain the moment generating function,probability generating function,cumulant generating function and characteristic function of probability distributions e.g beta distribution etc

Originally Posted by helpme2007

can someone pls tell me the steps or ideas or materials or hints on how to obtain the moment generating function,probability generating function,cumulant generating function and characteristic function of probability distributions e.g beta distribution etc

there are two ways:

i. Type the relevant terms into wikipedia and perform the given operations

ii. Get a suitable handbook in which they are tabulated (Abramowitz and
Stegun is probably a good place to start.

RonL

I like "Probability and Random Processes" by Grimmett and Stirzaker Clarendon press, Oxford, United Kindom. Lots of examples from different areas plus main ideas like moment generating functions, Laplace transform and Fourier.

Characteristic fcn is just probabilists fancy name for Fourier or Laplace transform or some discrete version of this. You use it to calc expected values. This is kind of basic idea. For details see Grimmett & Stirzaker, GS from now.

Answer to your other questions can also be found in this book:

Probability genr fcn used for non-negative RV say X is G(s)=E[s^X]. GS p.129

Moment genr fcn M(t)= E[exp t*X] sort of like Laplace transform. You use it to find EX = M'(0) where prim means derivative of M at point t=0. Higher moments EX^k = M^k(0), derivative order k. GS p.162

Cumulant generating fcn is the log of the moment generating fcn of RV say X.
log E[exp theta*X]. GS p.166

Good luck in calculating the char fcn for a RV with beta distribution..... Looks tough to me!