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and let X and S be sample mean be a random sample from N(u,0) 1. Let...
and let X and S be sample mean be a random sample from N(u,0) 1. Let are independent, follow the and sample variance, respectively. In order to show that X and S steps below X x-x2 , and show the joint pdf of 1-1) Use the change of variable technique X,X,,X n is (n 1s 202 1 f(F,x,) = n exp 202 a27 [Hint 1] Use Jacobian for n x n variable transformation [Hint 2] 4AT-r- des dis Je ddi...
Only 1-3) ,X, be a random sample from N(u,0") and let X and S be sample 1. Let mean and sample variance, respectively. In order to show that X and S are independent, tollow the steps below. x - x -X, and show the joint pdf of ,X,,..., X 1-1) Use the change of variable technique is (n-1)s n-u) еxp f(X,x 2a 20 av2n Use Jacobian for n x n variable transformation 1-2) Use the fact that X~N(4, /n), and...
Only 1-3) ,X, be a random sample from N(u,0") and let X and S be sample 1. Let mean and sample variance, respectively. In order to show that X and S are independent, tollow the steps below. x - x -X, and show the joint pdf of ,X,,..., X 1-1) Use the change of variable technique is (n-1)s n-u) еxp f(X,x 2a 20 av2n Use Jacobian for n x n variable transformation 1-2) Use the fact that X~N(4, /n), and...
Only 1-4) X, be a random sample from N(4,a ), , and let X and S be sample mean and sample 1. Let variance, respectively. In Order to show that and S are independent, tollow the steps below. and show the joint pdf of X,X3,*, X 1-1) Use the change of variable technique = Nx = x - is (п-1)5? п(т-и? f(E,x,) еxp ov2x 2a2 Use Jacobian for n x n variable transformation 1-2) Use the fact that X~N(u,a n)...
Only 1-6) N(4,) "x.xx be a random sample from variance, respectively. In order to show that and let X and S be sample mean and sample 1. Let and 5 are independent, tollow the steps below. 1-1) Use the change of variable technique =nx-x,- x and show the joint pdf of ,X,,X is (n-1) n- exp f(,x) 20 2a av2 Use Jacobian for n x n variable transformation 1-2) Use the fact that N(u,a n), and show that the conditional...
Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the distribution having p.d.f f(x) = e-y , 0<y<, zero elsewhere. Answer the following questions. (a) decide whether Z1 = Y2 and Z2=Y4-Y2 are stochastically independent or not. (hint. first find the joint p.d.f. of Y2 and Y4) (b) show that Z1 = nY1, Z2= (n-1)(Y2-Y1), Z3=(n-2)(Y3-Y2), ...., Zn=Yn-Yn-1 are stocahstically independent and that each Zi has the exponential distribution.(hint use change of variable technique)
Let X1,.. ,X be a random sample from an N(p,02) distribution, where both and o are unknown. You will use the following facts for this ques- tion: Fact 1: The N(u,) pdf is J(rp. σ)- exp Fact 2 If X,x, is a random sample from a distribution with pdf of the form I-8, f( 0,0) = for specified fo, then we call and 82 > 0 location-scale parameters and (6,-0)/ is a pivotal quantity for 8, where 6, and ô,...
3.4 Let X,, X be a random sample of size n from the U(Q,62) distribution, 6, and let Y, and Yn be the smallest and the largest order statistics of the Xs (i) Use formulas (28) and (29) in Chapter 6 to obtain the p.d.f.'s of Y and Y and then, by calculating depending only on Yi and 1,- Part i. (Note: it is not saying to find the joint pdf of Yi and Yn Find their marginal Theorem 13...
Question 2 Let X Pareto(r, 8 = 1) which has pdf: f(x) = 1 , 1 >1 and r > 1 (a) Given a random sample of size n from X show that the mle for r is: r* = 1/7 where Y = SEY and Y = log X (b) Let Y = log X Use the mgf technique (with t <r) to show that: Y Exp(1 = r) [ HINT: My(t) = Eletbox] = E[X“) = * **f(x)dt...
Let X, X,, ..., X, denote a random sample of size n from a population with pdf (10) = b exp(@m()).0<x<1 where (<O<0. Derive that the likelihood ratio test of H.:0=1 versus H, :0 #1 in terms of T(x) = ŽI (3)