Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

76 Probability and Distributions

1.9.7.Show that the moment generating function of the random variableXhaving

the pdff(x)=^13 ,− 1 <x<2, zero elsewhere, is

M(t)=

{ e 2 t−e−t

3 t t^ =0

1 t=0.

1.9.8.LetXbe a random variable such thatE[(X−b)^2 ] exists for all realb. Show
thatE[(X−b)^2 ] is a minimum whenb=E(X).

1.9.9.LetXbe a random variable of the continuous type that has pdff(x). Ifm

is the unique median of the distribution ofXandbis a real constant, show that

E(|X−b|)=E(|X−m|)+2

∫b

m

(b−x)f(x)dx,

provided that the expectations exist. For what value ofbisE(|X−b|) a minimum?

1.9.10.LetXdenote a random variable for whichE[(X−a)^2 ] exists. Give an
example of a distribution of a discrete type such that this expectation is zero. Such
a distribution is called adegenerate distribution.

1.9.11. LetX denote a random variable such thatK(t)=E(tX) exists for all
real values oftin a certain open interval that includes the pointt= 1. Show that
K(m)(1) is equal to themthfactorial momentE[X(X−1)···(X−m+1)].

1.9.12.LetXbe a random variable. Ifmis a positive integer, the expectation
E[(X−b)m], if it exists, is called themth moment of the distribution about the
pointb. Let the first, second, and third moments of the distribution about the point
7 be 3, 11, and 15, respectively. Determine the meanμofX, and then find the
first, second, and third moments of the distribution about the pointμ.

1.9.13.LetXbe a random variable such thatR(t)=E(et(X−b))existsfortsuch
that−h<t<h.Ifmis a positive integer, show thatR(m)(0) is equal to themth
moment of the distribution about the pointb.

1.9.14.LetXbe a random variable with meanμand varianceσ^2 such that the
third momentE[(X−μ)^3 ] about the vertical line throughμexists. The value of
the ratioE[(X−μ)^3 ]/σ^3 is often used as a measure ofskewness.Grapheachof
the following probability density functions and show that this measure is negative,
zero, and positive for these respective distributions (which are said to be skewed to
the left, not skewed, and skewed to the right, respectively).

(a)f(x)=(x+1)/ 2 ,− 1 <x<1, zero elsewhere.

(b)f(x)=^12 ,− 1 <x<1, zero elsewhere.

(c)f(x)=(1−x)/ 2 ,− 1 <x<1, zero elsewhere.
1.9.15.LetXbe a random variable with meanμand varianceσ^2 such that the
fourth momentE[(X−μ)^4 ] exists. The value of the ratioE[(X−μ)^4 ]/σ^4 is often
used as a measure ofkurtosis. Graph each of the following probability density
functions and show that this measure is smaller for the first distribution.