Let X1,..., X., be a independent identically distributed sample from the DISCRETE uniform distribution with probability...
Suppose X1, X2, ..., Xn are independent and identically distributed (iid) with a Uniform -0,0 distri- bution for some unknown e > 0, i.e., the Xi's have pdf Suppose X1, X2,..., Xn are independent and identically distributed (iid f(3) = S 20, if –0 < x < 0; 20 0, otherwise. (a) (4 pts) Briefly explain why or why not this is an exponential family (b) (5 pts) Find one meaningful sufficient statistic for 0. (By "meaningful”, I mean it...
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.
13. Let X1, X2, ...,Xy be a sequence of independent and identically distributed discrete random variables, each with probability mass function P(X = k)=,, for k = 0,1,2,3,.... emak (a) Find the expected value and the variance of the sample mean as = N&i=1X,. (b) Find the probability mass function of X. (c) Find an approximate pdf of X when N is very large (N −0).
3. Let X1, X2, . . . , Xn be independent samples of a random variable with the probability density function (PDF): fX(x) = θ(x − 1/ 2 ) + 1, 0 ≤ x ≤ 1 ,0 otherwise where θ ∈ [−2, 2] is an unknown parameter. We define the estimator ˆθn = 12X − 6 to estimate θ. (a) Is ˆθn an unbiased estimator of θ? (b) Is ˆθn a consistent estimator of θ? (c) Find the mean squared...
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...
Question 6 [15 marks] Let X1, X2,..., Xn be independent and identically distributed random vari- ables with common probability function ()p(1-p) m m-a ; x 0,1,. ., m otherwise 0 where m is known and p is unknown (a) Obtain the Sequential Probability Ratio Test of Ho p = po versus HA p P, where pi > po, with significance level 0.01 and power 0.95. Describe the test precisely; (b) For the case where po 3/8,pı = 1/2, m =...
2. (15pts) Let X1, X2 be independent and identically distributed with Uniform(0,) density. (a) Is Y-X1 + X2 a sufficient statistic for θ? Hint: You need to find the conditional density of (X1, X2) given Y = X1 + X2. (b) Consider now S := max(X1, X2). 1s S a sufficient statistics for θ?
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)...
Observations X1,..., Xn are independent identically distributed, following the PDF fx:(xi) = 0x8-1, and that 0<Xi <1 for all i. The parameter is an unknown positive number. Find the ML estimator of e
3. (a) (5 points) Let Xi,... be a sequence of independent identically distributed random variables e of tnduqendent idente onm the interval (o, 1] and let Compute the (almost surely) limit of Yn (b) (5 points) Let X1, X2,... be independent randon variables such that Xn is a discrete random variable uniform on the set {1, 2, . . . , n + 1]. Let Yn = min(X1,X2, . . . , Xn} be the smallest value among Xj,Xn. Show...