Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

5.2. Convergence in Distribution 339

Sample Size 5

y

Density

0246

0.0

0.1

0.2

0.3

0.4

0.5

Sample Size 10

y

Density

− 2 − 101234

0.0

0.1

0.2

0.3

0.4

Sample Size 20

y

Density

− 2 − 10123

0.00.10.20.30.4

Sample Size 50

y

Density

− 3 − 2 −10 1 2 3

0.0

0.1

0.2

0.3

0.4

Figure 5.2.2: For each value ofn, a histogram plot of 1000 generated values
ynis shown, whereynis discussed in Example 5.2.7. The limitingN(0,1) pdf is
superimposed on the histogram.

5.2.3.LetYndenote the maximum of a random sample of sizenfrom a distribution
of the continuous type that has cdfF(x)andpdff(x)=F′(x). Find the limiting
distribution ofZn=n[1−F(Yn)].

5.2.4.LetY 2 denote the second smallest item of a random sample of sizenfrom a
distribution of the continuous type that has cdfF(x)andpdff(x)=F′(x). Find
the limiting distribution ofWn=nF(Y 2 ).

5.2.5. Let the pmf ofYnbepn(y)=1,y=n, zero elsewhere. Show thatYn
does not have a limiting distribution. (In this case, the probability has “escaped”
to infinity.)

5.2.6.LetX 1 ,X 2 ,...,Xnbe a random sample of sizenfrom a distribution that is
N(μ, σ^2 ), whereσ^2 >0. Show that the sumZn=

∑n

1 Xidoes not have a limiting

distribution.

5.2.7.LetXnhave a gamma distribution with parameterα=nandβ,whereβis
not a function ofn.LetYn=Xn/n. Find the limiting distribution ofYn.

5.2.8.LetZnbeχ^2 (n)andletWn=Zn/n^2. Find the limiting distribution ofWn.

5.2.9.LetXbeχ^2 (50). Using the limiting distribution discussed in Example 5.2.7,
approximateP(40<X<60). Compare your answer with that calculated by R.