70 Probability and DistributionsExample 1.9.3.IfXhas the pdff(x)={ 1
x^21 <x<∞
0elsewhere,then the mean value ofXdoes not exist, because
∫∞1|x|1
x^2
dx= lim
b→∞∫b11
x
dx= lim
b→∞
(logb−log 1) =∞,which is not finite.We next define a third special expectation.Definition 1.9.3 (Moment Generating Function). Let X be a random variable
such that for some h> 0 , the expectation ofetXexists for−h<t<h.The
moment generating function ofXis defined to be the functionM(t)=E(etX),
for−h<t<h. We use the abbreviationmgfto denote the moment generating
function of a random variable.
Actually, all that is needed is that the mgf exists in an open neighborhood of 0.
Such an interval, of course, includes an interval of the form (−h, h)forsomeh>0.
Further, it is evident that if we sett=0,wehaveM(0) = 1. But note that for an
mgf to exist, it must exist in an open interval about 0.
Example 1.9.4.Suppose we have a fair spinner with the numbers 1, 2 ,and 3 on
it. LetXbe the number of spins until the first 3 occurs. Assuming that the spins
are independent, the pmf ofXis
p(x)=1
3(
2
3)x− 1
,x=1, 2 , 3 ,....Then, using the geometric series, the mgf ofXisM(t)=E(etX)=∑∞x=1etx1
3(
2
3)x− 1
=1
3
et∑∞x=1(
et2
3)x− 1
=1
3
et(
1 −et2
3)− 1
,provided thatet(2/3)<1; i.e.,t<log(3/2). This last interval is an open interval
of 0; hence, the mgf ofXexists and is given in the final line of the above derivation.
If we are discussing several random variables, it is often useful to subscriptM
asMXto denote that this is the mgf ofX.
LetXandY be two random variables with mgfs. IfXandY have the same
distribution, i.e, FX(z)=FY(z) for allz, then certainlyMX(t)=MY(t)ina
neighborhood of 0. But one of the most important properties of mgfs is that the
converse of this statement is true too. That is, mgfs uniquely identify distributions.
We state this as a theorem. The proof of this converse, though, is beyond the scope
of this text; see Chung (1974). We verify it for a discrete situation.