5.3. Central Limit Theorem 3415.2.19.LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample (see
Section 5.2) from a distribution with pdff(x)=e−x, 0 <x<∞,zero elsewhere.
Determine the limiting distribution ofZn=(Yn−logn).
5.2.20.LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample (see
Section 5.2) from a distribution with pdff(x)=5x^4 , 0 <x< 1 ,zero elsewhere.
Findpso thatZn=npY 1 converges in distribution.5.2.21.Consider Stirling’s formula (5.2.2):(a)Run the following R code to check this formuala fork=5tok= 15.
ks = 5; kstp = 15; coll = c();for(j in ks:kstp){
c1=gamma(j+1); c2=sqrt(2*pi)*exp(-j+(j+.5)*log(j))
coll=rbind(coll,c(j,c1,c2))}; coll(b)Take the log of Stirling’s formula and compare it with the R computation
lgamma(k+1).(c)Use Stirling’s formula to show that the first limit in Example 5.2.3 is 1.5.3 CentralLimitTheorem
It was seen in Section 3.4 that ifX 1 ,X 2 ,...,Xnis a random sample from a normal
distribution with meanμand varianceσ^2 , the random variable
∑n
i=1Xi−nμ
σ
√
n=√
n(Xn−μ)
σis, for every positive integern, normally distributed with zero mean and unit vari-
ance. In probability theory there is a very elegant theorem called theCentral
Limit Theorem(CLT). A special case of this theorem asserts the remarkable and
important fact that ifX 1 ,X 2 ,...,Xndenote the observations of a random sample
of sizenfrom any distribution having finite varianceσ^2 >0 (and hence finite mean
μ), then the random variable
√
n(Xn−μ)/σconverges in distribution to a random
variable having a standard normal distribution. Thus, whenever the conditions of
the theorem are satisfied, for largenthe random variable
√
n(Xn−μ)/σhas an
approximate normal distribution with mean zero and variance 1. It is then possible
to use this approximate normal distribution to compute approximate probabilities
concerningX.
We often use the notation “Ynhas a limiting standard normal distribution” to
mean thatYnconverges in distribution to a standard normal random variable; see
Remark 5.2.1.
The more general form of the theorem is stated, but it is proved only in the
modified case. However, this is exactly the proof of the theorem that would be
given if we could use the characteristic function in place of the mgf.