2. (15pts) Let X1, X2 be independent and identically distributed with Uniform(0,) density. (a) Is Y-X1...
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...
2) Let Yİ,Ý,, ,y, be independent and identically distributed from the distribution with density where c > 0 is a constant and θ > 0. Find the MLE for 60. 2) Let Yİ,Ý,, ,y, be independent and identically distributed from the distribution with density where c > 0 is a constant and θ > 0. Find the MLE for 60.
18. Let X, X2, ..., Xv are independent and identically distributed standard uniform random variables. Find the following expectations: (a) E[max(X,,X2, .Xn,)] (b) E[min(X1,X2,..., Xn)]
3. Let {X1, X2, X3, X4} be independent, identically distributed random variables with p.d.f. f(0) = 2. o if 0<x< 1 else Find EY] where Y = min{X1, X2, X3, X4}.
Let (X1, Y1) and (X2, Y2) be independent and identically distributed continuous bivariate random variables with joint probability density function: fX,Y (x,y) = e-y, 0 <x<y< ; =0 , elsewhere. Evaluate P( X2>X1, Y2>Y1) + P (X2 <X1, Y2<Y1) .
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.
4. Let X1,..., Xn be independent, identically distributed random vari- ables with common density 2 log c)? f(0; 1) = 0<<1, XCV21 (>0). : 212 (a) Find the form of the critical region C'* for the most powerful test of H:/= 1 vs. HQ: >1. (b) Suppose the n = 20 and a = .10. Find the specific value for the cutoff value) K from the critical region C* in part (a). (Hint: Show that Y = (log X/X) is...
NB: Please do it for Let X1, X2, ;;;;, Xn are independent and Not identiically instead of identically Not identically Let X ,X2, ..., Xy are independent and Identically distributed standard uniform random variables. Find the following expectations: (a) E[max(X1, X2, ...,XN)] (b) E[min(X1, X2, ...,Xy)]
10. (8 marks) Suppose Y, Y is a random sample of independent and identically distributed random variables with density function given by else a) (5 marks) By conditioning (definition 9.3) show that Uis sufficient for 0 b) (3 marks) By factorization (theorem 9.4) show that U- is sufficient for 0 Definition 9.3: Sufficiency The statistic U-g(, , X,) is said to be sufficient for θ if the conditional distribution of Y, Y, given U, does not depend on e Theorem...