Exercise 2 (Monte Carlo integration). Let (Xk)kzl be i.i.d. Uniform([0, 1]) RVs and let f: [0,1]...
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.
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.
number 3 please Hw4.1708.pd 1 2 TL (2) LP convergence vs. convergence in probability Let Xn, nNbe a sequence of random variables and let X be another random variable. Given l < p < oo, we say that Xn converges to X in Lp if E(Xn-X") → 0 as n → x Show that this implies that Xn converges to X in probability (3) Monte Carlo Let f : 10, 1] → R be continuous and let Xn, n on...
0, oo) which converges to a certain real Let f be a real-valued continuous function over o0, i.e., lim f(x) = A. Answer the following questions value A as Find the following limit lim aoo a2 f (x)dx 0, oo) which converges to a certain real Let f be a real-valued continuous function over o0, i.e., lim f(x) = A. Answer the following questions value A as Find the following limit lim aoo a2 f (x)dx
Gice excel formula please 4. Estimate f x2 dx between 0 and 2 using Monte Carlo Inte ratio on. Here's how: Generate 1,000 random (x.y) pairs where 0
Suppose f is a continuous and differentiable function on [0,1] and f(0)= f(1). Let a E (0, 1). Suppose Vr,y(0,1) IF f'(x) 0 and f'(y) ±0 THEN f'(x) af'(y) Show that there is exactly f(ax) and f'(x) 0 such that f(x) one Hint: Suppose f(x) is a continuous function on [0, 1] and f(0) x € (0, 1) such that f(x) = f(ax) f(1). Let a e (0,1), there exists an Suppose f is a continuous and differentiable function on...
Let X1....Xn be i.i.d sample with a continous distribution function F(.) and X(1)<......<X(n) are the orser-statistics of the sample. Let the constant Mp be defined by F(Mp)=p. Show that for 1≤k1≤k2≤ n, P{X(k1) ≤Mp ≤X(k2)}=P{k1 ≤Bionmial(n,p) k2}
Let X1. . . . Xn be i.i.d Uniform over the interval (θ, θ + 1].Show that X(1)+X(n) )/2- 1/2 is also an unbiased estimator of θ, whereX(1) is the minimum order statistic and X(n) is the maximum order statistic. If X - 1/2 is also an unbiased estimator of θ which of the two estimators would you prefer to use.
Problem 5 Let f : [0,1] → R be continuous and assume f(zje (0, 1) for all x E (0,1). Let n E N with n 22. Show that there is eractly one solution in (0,1) for the equation 7L IC nx+f" (t) dt-n-f(t) dt.
Problem 2. (The Convergence of Extreme Value) Let X1, X2, ... be i.i.d sample from the distribution with density function as: f(x) = >1 10 otherwise Define Mn = min(X1, X2, ... , Xn), answer the following questions. 1) Show that Mn P 1 as n +0. 2) Show that n(Mn – 1) converges in distribution as n + 00. Find out the limit distri- bution.