Question

Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order statistic (b) Find the limiting distribution of XT (c) Find the limiting distribution of n In XIm 3. Suppoe that Xi,x ,x, are i.id. N(0, σ2). Find a function of T(X)-Σ-1 X? that con- verges in distribution to a normal distribution. State the mean and variance of your limiting normal distribution. 4. Stirling's Formula, which gives approximation for factorials, can be derived using CLT (a) Suppose that Xi, X2,.,Xn is an ii.d. sample from Exp(1). Show that, for a standard normal random variable Z, (b) Show Vn by differencing both sides of the approximation in part a. Then set r-0 to get Stirling's Formula. 5. Suppose that Y, is an tid sample from Negative Binomial (a.p). Give a normal approximation of use CLT, when n is large. 6. (Mandatory for Graduate Student. Extra credit for undergrad.) Let Xi,X,..Xn be a iid. sample from Bernoulli(p) and let Y, - EX-P)/n. Show that Y converges to a degenerate distribution at 0 as n -oo.

Answer 1