Please answer all the parts neatly with all details.
Please answer all the parts neatly with all details. 8. Let X1, X2, ... be an...
Please answer all the parts neatly with all details.
3. Assume X1, X2,... are a sequence of i.i.d. random variables having finite first moment, that is, v = E(Xi oo. Let Yn = (|X1| .+ |Xn|)/n. (a) Show that Yn ->v in probability. (b) Show that E(Y,) -- v. (c) Show that E(|X, - /u|) -0 where u = E(X)
3. Assume X1, X2,... are a sequence of i.i.d. random variables having finite first moment, that is, v = E(Xi...
Please answer all the parts neatly with all details.
4. Let X1, X2,... be independent random variables satisfying E(X4) < B for some finite B > 0 and E(Xn)-> . (a) Show that Y = Xn - E(Xn) are independent and E(Yn) = 0, E(Y2) < B, E(Y4 (b) Show that for Y, = (Yi +..+ Y)/n, 16B 16B ΣειβΥ< 6B 1 = n4 i=1 6 + n4 ij ΣΕ Υ) E(Y4) n'3 n2 P(Y > €) < oo and...
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 → ∞.
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.
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following convergence results independently (i.e, do not conclude the weaker convergence modes from the stronger ones). d Yn 0. a. P b.Y 0. L 0, for all r 1 Yn C. a.s d. Y 0.
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following...
suppose X1 -> Xn is a random sample from a uniform distribution on the interval [0,theta]. let X1 = min {X1,X2,...Xn} and let Yn= nX1. show that Yn converges in distribution to an exponential random variable with mean theta.
Central Limit Theorem: let x1,x2,...,xn be I.I.D. random variables with E(xi)= U Var(xi)= (sigma)^2 defind Z= x1+x2+...+xn the distribution of Z converges to a gaussian distribution P(Z<=z)=1-Q((z-Uz)/(sigma)^2) Use MATLAB to prove the central limit theorem. To achieve this, you will need to generate N random variables (I.I.D. with the distribution of your choice) and show that the distribution of the sum approaches a Guassian distribution. Plot the distribution and matlab code. Hint: you may find the hist() function helpful
5. (4 points) Let X1, X2, be independent random variables that are uniformly distributed on [-1,1] Show that the sequence Yi,Y2, converges in probability to some limit, and identity the limit, for each of the following cases: (a) Yn = max Xi, , xn (this is similar to an example from class). (c) Yn = (Xn)"
Please answer all the parts neatly with all details.
6. Two independent random variables X and Y have the same distribution with finite second moment Assume X and (X +Y)/2 have the same distribution (a) Show that the expectation of X is zero. (b) Show that X and (X1 ...+X2n)/V2n have the same distribution where X\, X2,... are independent and have identical distribution of X (c) Show that X~ N(0, 02) for some o2 > 0.
6. Two independent random...
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.