Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

1.9. Some Special Expectations 77

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

(b)f(x)=3(1−x^2 )/ 4 ,− 1 <x<1, zero elsewhere.

1.9.16.Let the random variableXhave pmf

p(x)=

⎧

⎨

⎩

px=− 1 , 1

1 − 2 px=0

0elsewhere,

where 0<p<^12. Find the measure of kurtosis as a function ofp. Determine its

value whenp=^13 ,p=^15 ,p= 101 ,andp= 1001. Note that the kurtosis increases as

pdecreases.

1.9.17.Letψ(t)=logM(t), whereM(t) is the mgf of a distribution. Prove that
ψ′(0) =μandψ′′(0) =σ^2. The functionψ(t) is called thecumulant generating
function.

1.9.18.Find the mean and the variance of the distribution that has the cdf

F(x)=

⎧

⎪⎪

⎨

⎪⎪

⎩

0 x< 0

x

8 0 ≤x<^2

x^2

16 2 ≤x<^4

14 ≤x.

1.9.19.Find the moments of the distribution that has mgfM(t)=(1−t)−^3 ,t<1.

Hint:Find the Maclaurin series forM(t).

1.9.20.We say thatXhas aLaplacedistribution if its pdf is

f(t)=

1

2

e−|t|, −∞<t<∞. (1.9.5)

(a)Show that the mgf ofXisM(t)=(1−t^2 )−^1 for|t|<1.

(b)ExpandM(t) into a Maclaurin series and use it to find all the moments ofX.

1.9.21.LetXbe a random variable of the continuous type with pdff(x), which
is positive provided 0<x<b<∞, and is equal to zero elsewhere. Show that

E(X)=

∫b

0

[1−F(x)]dx,

whereF(x)isthecdfofX.

1.9.22. LetX be a random variable of the discrete type with pmfp(x)thatis
positive on the nonnegative integers and is equal to zero elsewhere. Show that

E(X)=

∑∞

x=0

[1−F(x)],

whereF(x)isthecdfofX.