556 MATH - EXERCISES 5 These exercises are not for assessment
1. Using the Central Limit Theorem, construct Normal approximations to probability distribution of a random variable X having (i) (ii) (iii) (iv) a Binomial distribution, X Binomial(n, ) a Poisson distribution, X P oisson() a Negative Binomial distribution, X N egBinomial(n, ) a Gamma distribution,X Gamma(, )
In the following questions, use the following results concerning extreme order statistics; let Yn and Zn correspond to the maximum and minimum order statistics derived from random sample X1 , ...Xn from population with cdf FX , that is Yn = max {X1 , ..., Xn } Then the cdfs of Yn and Zn are given by FYn (y) = {FX (y)}n 2. Suppose X1 , ..., Xn U nif orm(0, 1), that is FX (x) = x 0x1 FZn (z) = 1 {1 FX (z)}n . Zn = min {X1 , ..., Xn } .
Find the cdfs of Yn and Zn , and the limiting distributions as n . 3. Suppose X1 , ..., Xn have cdf FX (x) = 1 x1 x1
n Find the cdfs of Zn and Un = Zn , and the limiting distributions of Zn and Un as n .
4.
Suppose X1 , ..., Xn have cdf
1 xR 1 + ex Find the cdfs of Yn and Un = Yn log n and the limiting distributions of Yn and Un as n . FX (x) = 1 x>0 1 + x Find the cdfs of Yn and Zn , and the limiting distributions as n . Find also the cdfs of Un = Yn /n and Vn = nZn , and the limiting distributions of Un and Vn as n . FX (x) = 1 Suppose X1 , ..., Xn P oisson() are independent random variables. Let 1 Mn = n
p n
5.
Suppose X1 , ..., Xn have cdf
6.
Xi
i=1
Show Mn that as n . If random variable Tn is dened by Tn = eMn , show that p Tn e , and nd an approximation to the probability distribution of Tn as n .
MATH 556 EXERCISES 5
Page 1 of 2
7.
For the following sequences of random variables, {Xn }, decide whether the the sequence converges in mean-square (rth mean for r = 2) or in probability as n . 1 with prob. 1/n (a) Xn = 2 with prob. 1 1/n n2 with prob. 1/n (b) Xn = 1 with prob. 1 1/n n with prob. 1/ log n (c) Xn = 0 with prob. 1 1/ log n
Almost sure convergence and the Borel-Cantelli Lemma. 8. Consider the sequence of random variables dened for n = 1, 2, 3, ... by Xn = I[0,n1 ) (Un ) where U1 , U2 , ... are a sequence of independent U nif orm(0, 1) random variables, and IA is the indicator function for set A 1 A IA () = 0 A / Does the sequence {Xn } converge (a) (b) almost surely ? in rth mean for r = 1 ?
[Hint: ...