Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

78 Probability and Distributions

1.9.23.LetXhave the pmfp(x)=1/k, x=1, 2 , 3 ,...,k, zero elsewhere. Show

that the mgf is

M(t)=

{

et(1−ekt)

k(1−et) t^ =0

1 t=0.

1.9.24.LetXhave the cdfF(x) that is a mixture of the continuous and discrete
types, namely

F(x)=

⎧

⎨

⎩

0 x< 0
x+1
4 0 ≤x<^1
11 ≤x.
Determine reasonable definitions ofμ=E(X)andσ^2 =var(X) and compute each.
Hint: Determine the parts of the pmf and the pdf associated with each of the
discrete and continuous parts, and then sum for the discrete part and integrate for
the continuous part.

1.9.25.Considerkcontinuous-type distributions with the following characteristics:
pdffi(x), meanμi,andvarianceσ^2 i,i=1, 2 ,...,k.Ifci≥ 0 ,i=1, 2 ,...,k,and
c 1 +c 2 +···+ck= 1, show that the mean and the variance of the distribution having
pdfc 1 f 1 (x)+···+ckfk(x)areμ=

∑k

i=1ciμiandσ

(^2) =∑k
i=1ci[σ
2
i+(μi−μ)
(^2) ],
respectively.
1.9.26.LetXbe a random variable with a pdff(x)andmgfM(t). Supposefis
symmetric about 0; i.e.,f(−x)=f(x). Show thatM(−t)=M(t).
1.9.27.LetXhave the exponential pdf,f(x)=β−^1 exp{−x/β},0<x<∞, zero
elsewhere. Find the mgf, the mean, and the variance ofX.

1.10ImportantInequalities

In this section, we discuss some famous inequalities involving expectations. We
make use of these inequalities in the remainder of the text. We begin with a useful
result.

Theorem 1.10.1. LetXbe a random variable and letmbe a positive integer.
SupposeE[Xm]exists. Ifkis a positive integer andk≤m,thenE[Xk]exists.

Proof:We prove it for the continuous case; but the proof is similar for the discrete
case if we replace integrals by sums. Letf(x)bethepdfofX.Then

∫∞

−∞

|x|kf(x)dx =

∫

|x|≤ 1

|x|kf(x)dx+

∫

|x|> 1

|x|kf(x)dx

≤

∫

|x|≤ 1

f(x)dx+

∫

|x|> 1

|x|mf(x)dx

≤

∫∞

−∞

f(x)dx+

∫∞

−∞

|x|mf(x)dx

≤ 1+E[|X|m]<∞, (1.10.1)