(a) Suppose that i, X2,... , In is an i.i.d. sample from Exp(1). Show that, for...
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...
Let X1, X2, · · · Xn be a i.i.d. sample from Bernoulli(p) and let . Show that Yn converges to a degenerate distribution at 0 as n → ∞.
3. Suppose that X1,X2, ,Xn are i.id. N(0, σ2). Find a function of T(X)-Σǐii verges in distribution to a normal distribution. State the mean and variance of your limiung normal distribution. 4. Stirling's Formula, which gives approximation for factorials, can be derived using CLT. (a) Suppose that X1, X2, random variable Z, .Xn is an ii.d. sample from Exp(1). Show that, for a standard normal PTPZ) (b) Show by differencing both sides of the approximation in part a. Then set...
Let Xi, X2...-Xn be a iid. sample from Bernoulli(p) and let Yn-Σηι(X-P)/n. Show that Ya converges to a degenerate distribution at 0 as n-o.
Stirling’s Formula, which gives approximation for factorials, can be derived using CLT. (a) Suppose that X1, X2, · · · , Xn is an i.i.d. sample from Exp(1). Show that, for a standard normal random variable Z, (b) Show by differencing both sides of the approximation in part a. Then set x = 0 to get Stirling’s Formula. We were unable to transcribe this imageГ(n) уж
tirling's Formula, which gives approximation for factorials, can be derived using CLT. (a) Suppose that X1, X2, random variable Z, , Xn is an ii.d. sample from Exp(1). Show that, for a standard normal (b) Show by differencing both sides of the approximation in part a. Then set 0 to get Stirling's Formula.
Exercise 5.23. Let (Xn)nz1 be a sequence of i.i.d. Bernoulli(p) RVs. Let Sn -Xi+Xn (i) Let Zn-(Sn-np)/ V np (1-p). Show that as n oo, Zn converges to the standard normal RV Z~ N(0,1) in distribution. (ii) Conclude that if Yn~Binomial(n, p), then (iii) From i, deduce that have the following approximation x-np which becomes more accurate as n → oo.
(b) For n = 100, give an approximaation for P(Y> 100) (c) Let X be the sample mean, then approximate P(1.1< 1.2) for -100. 2. Consider a random sample XX from CDF F(a) 1-1/ for z [1, 0o) and zero otherwise. (a) Find the limiting distribution of XiI.n, the smallest order statistic. (b) Find the limiting distribution of XI (c) Find the limiting distribution of n In X1:m- 3. Suppose that X,,, are iid. N(0,o2). Find a function of T(x)x...
Exercise 5.22. Let (Xn)nel be a sequence of i.i.d. Poisson(a) RVs. Let Sn-X1++Xn (i) Let Zn-(Sn-nA)/Vm. Show that as n-, oo, Zn converges to the standard normal RV Z ~ N(0,1) in distribution (ii) Conclude that if Yn~Poisson(nX), then ii) Fromii) deduce that we have the following approximation which becomes more accurate as noo.
1. Let Xi, X2, X, be a 1.1.d. sample form Exp(1), and Y = Σ=i Xi. (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(1.1 < X < 1.2) for n = 100.