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What is moment generating function?

1 Answer
Dec 31, 2017

Answer:

See below.

Explanation:

The parameters #mu# (mean) and #sigma# (standard deviation) locate the center and describe the spread associated with the values of a random variable #Y#. They do not, however, provide a unique characterization of the distribution of #Y#; many different distributions possess the same means and standard deviations.

  • The #k#th moment of a random variable #Y# taken about the origin is defined to be #E(Y^k)# and is denoted by #mu_k^'#

  • The moment-generating function #m(t)# for a random variable #Y# is defined to be #m(t)=E(e^(tY))#. We say that a moment-generating function for #Y# exists if there exists a positive constant #b# such that #m(t)# is finite for #abs(t) <= b#

If #m(t)# exists, then for any positive integer #k#,

#(d^km(t))/(dt^k)]_(t=0)=m^(k)(0)=mu_k^'#

In other words, if you find the #k#th derivative of #m(t)# with respect to #t# and then set #t=0#, the result will be #mu_k^'#.

Then we find that various probability distributions have their own unique moment-generating function.

Mathematical Statistics, Wackerly

Mathematical Statistics, Wackerly