4. Let Xi i = 1,2, ... be independent and identically distributed (i.i.d.) Possion. That is...
Problem 7. Let Xi, X2,..., Xn be i.i.d. (independent and identically distributed) random variables with unknown mean μ and variance σ2. In order to estimate μ and σ from the data we consider the follwing estimates n 1 Show that both these estimates are unbiased. That is, show that E(A)--μ and
Let X1, ..., Xn be i.i.d. [Recall that i.i.d. stands for independent and identically distributed.] Since X1, ..., Xn all have the same distribution, they have the same expected value and variance. Let E(X1) = µ and V ar(X1) = σ 2 . Find the following in terms of µ and σ 2 . (a) E(X2 1 ). Note this is not µ 2 ! (b) E( Pn i=1 X2 i /n). (c) Now, define W by W = 1...
(5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b) Find E(V) and E(W) (5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b)...
Exercise 7 (Ancilliarity) Choose one: 1. Let {X;} –1 be independent and identically distributed observations from a location paramter family with cumulative distribution function F(x – 0), -00 < 0 < 0. Show that range of the distribution of R = maxi(Xi) – mini(Xi) does not depend on the parameter 8.) Hint: Use the facts that X1 = Z1 + 0 , ..., Xin = Zn + 0 and mini(Xi) = mini(Zi + 0), maxi(Xi) = maxi(Z; +0), where {Zi}=1...
Let X1,X2,...,Xn be an independent and identically distributed (i.i.d.) random sample of Beta distribution with parameters α = 2 and β = 1, i.e., with probability density function fX(x) = 2x for x ∈ (0,1). Find the probability density function of the first and last order statistics Y1 and Yn.
8. Let X1...., X, be i.i.d. ~E(1) random variables (i.e., they are independent and identically distributed, all with the exponential distribution of parameter 1 = 1). a) Compute the cdf of Yn = min(X1,...,xn). b) How do P({Y, St}) and P({X1 <t}) compare when n is large and t is such that t<? c) Compute the odf of Zn = max(X1...., X.). d) How do P({Zn2 t}) and P({X1 2 t}) compare when n is large and t is such...
Let Xi,X2, , Xn be independent and identically distributed (ii.d.) Exponential(1) random variables. 14] [41 (a) Find the method of moments estimator for X (b) Find the method of moments estimator for (c) Find the bias, variance and MSE (mean square erop) for the essimator in part () Total: [16] Let Xi,X2, , Xn be independent and identically distributed (ii.d.) Exponential(1) random variables. 14] [41 (a) Find the method of moments estimator for X (b) Find the method of moments...
[PLEASE USE HINT] Problem 10: 10 points Suppose that (Xi, X2,... are independent identically distributed binary variables taking (0 1) values with probability PX here 0<q Introduce the new variable, M such that the event {M = k} occurs when three consecutive successes appear at the first time. In other words, event [M = kj, where k-3, occurs if and only if and there is no previous occurrence of three consecutive successes. Use conditioning techniques to derive the expected value,...
Let X and Y be two independent and identically distributed random variables that take only positive integer values. Their PMF is pX(n)=pY(n)=2−n for every n∈N , where N is the set of positive integers. Fix a t∈N . Find the probability P(min{X,Y}≤t) . Your answer should be a function of t . unanswered Find the probability P(X=Y) . unanswered Find the probability P(X>Y) . Hint: Use your answer to the previous part, and symmetry. unanswered Fix a positive integer k...
5. Let Xi i = 1,2, . ,N be i.i.d. U(0,1). Let Z = max{X1, .,Xn} and find Fz.