1.9. Some Special Expectations 75Every distribution has a unique characteristic function; and to each charac-
teristic function there corresponds a unique distribution of probability. IfXhas
a distribution with characteristic functionφ(t), then, for instance, ifE(X)and
E(X^2 ) exist, they are given, respectively, byiE(X)=φ′(0) andi^2 E(X^2 )=φ′′(0).
Readers who are familiar with complex-valued functions may writeφ(t)=M(it)
and, throughout this book, may prove certain theorems in complete generality.
Those who have studied Laplace and Fourier transforms note a similarity be-
tween these transforms andM(t)andφ(t); it is the uniqueness of these transforms
that allows us to assert the uniqueness of each of the moment-generating and char-
acteristic functions.
EXERCISES
1.9.1.Find the mean and variance, if they exist, of each of the following distribu-
tions.
(a)p(x)=x!(33!−x)!(^12 )^3 ,x=0, 1 , 2 ,3, zero elsewhere.(b)f(x)=6x(1−x), 0 <x<1, zero elsewhere.(c)f(x)=2/x^3 , 1 <x<∞, zero elsewhere.
1.9.2.Letp(x)=(^12 )x,x=1, 2 , 3 ,..., zero elsewhere, be the pmf of the random
variableX. Find the mgf, the mean, and the variance ofX.
1.9.3.For each of the following distributions, computeP(μ− 2 σ<X<μ+2σ).
(a)f(x)=6x(1−x), 0 <x<1, zero elsewhere.
(b)p(x)=(^12 )x,x=1, 2 , 3 ,..., zero elsewhere.
1.9.4.If the variance of the random variableXexists, show that
E(X^2 )≥[E(X)]^2.
1.9.5.Let a random variableX of the continuous type have a pdff(x)whose
graph is symmetric with respect tox=c.IfthemeanvalueofXexists, show that
E(X)=c.
Hint: Show thatE(X−c) equals zero by writingE(X−c)asthesumoftwo
integrals: one from−∞tocand the other fromcto∞. In the first, lety=c−x;
and, in the second,z=x−c. Finally, use the symmetry conditionf(c−y)=f(c+y)
in the first.
1.9.6. Let the random variableX have meanμ, standard deviationσ,andmgf
M(t),−h<t<h. Show that
E(
X−μ
σ)
=0,E[(
X−μ
σ) 2 ]
=1,and
E{
exp[
t(
X−μ
σ)]}
=e−μt/σM(
t
σ)
, −hσ<t<hσ.