Exercise 5.23. Let (Xn)nz1 be a sequence of i.i.d. Bernoulli(p) RVs. Let Sn -Xi+Xn (i) Let...
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.
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 → ∞.
(a) Suppose that i, X2,... , In is an i.i.d. sample from Exp(1). Show that, for a standard normal random variable Z b) Show Г(n) by differencing both sides of the approximation in part a. Then set a -0 to get Stirling's Formula. 5. Suppose that Y is an id sample from Negative Binomial (n,p). Give a normal approximation of Yn use CLT, when n is large. 6. (Mandatory for Graduate Student. Extra credit for undergrad.) Let Ai, converges to...
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.
5. Let Xi i = 1,2, . ,N be i.i.d. U(0,1). Let Z = max{X1, .,Xn} and find Fz.
Exercise 2 (Monte Carlo integration). Let (Xk)kzl be i.i.d. Uniform([0, 1]) RVs and let f: [0,1] -- R be a continuous function. For each n2 1, let (f(X)f(X2).+f(Xn)) (3) In = -- .. + Sof(x) dx in probability. (i) Suppose o f (x)| dx (ii) Further assume that f lf(x)2 dx <o0. Use Chebyshef's inequality to show that :< oo. Show that In P (IIn-I2 alVnVar(f(X1)) a2 f(x)2 dx (4)
Can someone help me with part (c), (with detailed explanation) Suppose that Xi,.. Xn are independent and identically distributed Bernoulli random variables, with mass function P (Xi = 1) = p and P (Xi = 0) = 1-p for some p (0,1) (a) For each fixed p є (0,1), apply the central limit theorem to obtain the asymptotic distribution of Σ.Xi, after appropriate centering and normalisation. (b) Derive the moment generating function of a Poisson(A) distribution. (c) Now suppose that...
Problem 3 Let Xi, X2,... , Xn be a sequence of binary, i.i.d. random variables. Assume P (Xi 1) P (Xi = 0) = 1/2. Let Z be a parity check on seluence Xi, X2, ,X,, that is, Z = X BX2 e (a) Is Z statistically independent of Xi? (Assume n> 1) (b) Are X, X2, ..., Xn 1, Z statistically independent? (c) Are X, X2,.., Xn, Z statistically independent? (d) Is Z statistically independent of Xi if P...
Problem 5 Let Xi, X2, ..., Xn be a random sample from Bernoulli(p), 0 < p < 1, and 7.i. Prove that the sample proportion is an unbiased estimator of p, i.e. p,- is an unbiased estimator of p 7.ii. Derive an expression for the variance of p,n 7.iii. Prove that the sample proportion is a consistent estimator of p. 7.iv. Prove that pn(1- Pn)
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...