1. Let X,.., Xn be independent and identically distributed as N (0,9) (here 9 is the...
1. Let X Xn be independent and identically distributed as N(0,9) (here 9 is the variance) (a) What is the distribution of Y-X,? (Verify it using MGF) (b) What is the distribution of Xn? (Again verify it using MGF) (c) Assume 25. What is the probability that an observed value of Xn lies inside the interval [-1.2,1.2 (d) Give a lower bound on the probability that Xn lies inside the interval [-1.2,12 using Chebyshev's inequality. Compare it with part (c)....
(1 point) In Unit 3, I claimed that the sum of independent, identically distributed exponential random variables is a gamma random variable. Now that we know about moment generating functions, we can prove it. Let X be exponential with mean A 4. The density is 4 a) Find the moment generating function of X, and evaluate at t 3.9 The mgf of a gamma is more tedious to find, so l'll give it to you here. Let W Gamma(n, A...
3. Let X , X, be an independent and identically distributed random sample from a distribution with pdf f(X; B) = B Xe 4X X20 a. Find the Maximum Likelihood Estimator of B, and denote it ß. b. Find the lower bound for Var() using Cramer-Rao inequality for ſunbiased.
Let X1...Xn be independent, identically distributed random sample from a poisson distribution with mean theta. a. Find the meximum liklihood estimator of theta, thetahat b. find the large sample distribution for (sqrt(n))*(thetahat-theta) c. Construct a large sample confidence interval for P(X=k; theta)
1. Let X1, X2,... , Xn be independent and identically distributed according to the unifornm distribution on (0,1). Let Xn and fn denote the 6th smallest and its pdf, respectively Determine fn(x) limn
Question 6 Let X1, . . . , Xn denote a sequence of independent and identically distributed i.id. N(14x, σ2) random variables, and let Yı, . . . , Yrn denote an independent sequence of iid. Nụy, σ2) ran- dom variables. il Λί and Y is an unbiased estimator of μ for any value of λ in the unit interval, i.e. 0 < λ < 1. 2. Verify that the variance of this estimator is minimised when and determine the...
Let X and Y be two independent and identically distributed random variables with expected value 1 and variance 2.56. First, find a non-trivial upper bound for P(|X + Y − 2| ≥ 1). Now suppose that X and Y are independent and identically distributed N(1,2.56) random variables. What is P(|X + Y − 2| ≥ 1) exactly? Why is the upper bound first obtained so different from the exact probability obtained?
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.
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...
1. The random variables Xi, X2,... are independent and identically distributed (iid), . .. are independent and identica each with pdf f given in Assignment 4, Question 1. Let s, X1 + . .. + Xn. Using the Central Limit Theorem and the graph of the standard normal distribution in Figure 1, approximate the probability P(S100 > 600). Express your answer in the format x.x - 10*. Verify your answer by simulating 10,000 outcomes of S1o0 and counting how many...