Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

348 Consistency and Limiting Distributions

5.3.9.Letf(x)=1/x^2 , 1 <x<∞, zero elsewhere, be the pdf of a random
variableX. Consider a random sample of size 72 from the distribution having this
pdf. Compute approximately the probability that more than 50 of the observations
of the random sample are less than 3.

5.3.10.Forty-eight measurements are recorded to several decimal places. Each of
these 48 numbers is rounded off to the nearest integer. The sum of the original 48
numbers is approximated by the sum of these integers. If we assume that the errors
made by rounding off are iid and have a uniform distribution over the interval
(−^12 ,^12 ), compute approximately the probability that the sum of the integers is
within two units of the true sum.

5.3.11.We know thatXis approximatelyN(μ, σ^2 /n)forlargen.Findtheap-

proximate distribution ofu(X)=X
3
, provided thatμ =0.

5.3.12.LetX 1 ,X 2 ,...,Xnbe a random sample from a Poisson distribution with
meanμ.Thus,Y=

∑n

i=1Xihas a Poisson distribution with meannμ.Moreover,

X=Y/nis approximatelyN(μ, μ/n)forlargen. Show thatu(Y/n)=

√

Y/nis a

function ofY/nwhose variance is essentially free ofμ.

5.3.13.Using the notation of Example 5.3.5, show that equation (5.3.4) is true.

5.3.14.Assume thatX 1 ,...,Xnis a random sample from a Γ(1,β) distribution.
Determine the asymptotic distribution of

√
n(X−β). Then find a transformation
g(X) whose asymptotic variance is free ofβ.

5.4 ∗ExtensionstoMultivariateDistributions

In this section, we briefly discuss asymptotic concepts for sequences of random
vectors. The concepts introduced for univariate random variables generalize in a
straightforward manner to the multivariate case. Our development is brief, and
the interested reader can consult more advanced texts for more depth; see Serfling
(1980).
We need some notation. For a vectorv∈Rp, recall the Euclidean norm ofvis
defined to be

‖v‖=

√√

√

√

∑p

i=1

vi^2. (5.4.1)

This norm satisfies the usual three properties given by

(a) For allv∈Rp,‖v‖≥0, and‖v‖=0ifandonlyifv= 0.

(b) For allv∈Rpanda∈R,‖av‖=|a|‖v‖.

(c) For allv,u∈Rp,‖u+v‖≤‖u‖+‖v‖.

(5.4.2)

Denote the standard basis ofRpby the vectorse 1 ,...,ep, where all the components
ofeiare 0 except for theith component, which is 1. Then we can write any vector