1. Suppose X1, X2, ..., Xn are independent and identically distributed random variables from1N(µ, o3) (each with a density f(x; µ) =206 ). Suppose that oo is known.V2n00a) Argue that the joint density of X1, X2, .X, has monotone likelihood ratio in E X;.b) Derive the UMP unbiased size a = 0.05 test y* of Ho : u = µo versus H1 : µ # µo.c) Show that the power function of this test isVn(µ – H0)Vn(u – Ho)Euf* = 1 – ¢1.96-1.96σοσοwith denoting the cdf of the standard normal distribution.d) Set n = 10, o0 = 2, µo = 3. Evaluate numerically the power function for u = 1, 2, 3, 4, 5and draw its graph on the real axis using R.e) Calculate the density fxe (x) of the third order statistic X(3) under Họ. Hence findnumerically P(X (3) < 2). (You could use the integrate function in R ).

Let consider X1, X2, ........, Xn are the independent and identically distributed random variables…

Suppose X1, X2, ..., Xn are independent and identically distributed random variables from N(µ, o3) (each with a density f (x; µ) = (z-µ)² 20ó ). Suppose that oo is known. e V2T00 a) Argue that the joint density of X1, X2,..., Xn has monotone likelihood ratio in E X;. i=D1 b) Derive the UMP unbiased size a = 0.05 test p* of Ho : u = µo versus H1 :µ # po. c) Show that the power function of this test is (Or1 – 1)u^ + +-1.96 Eµ4* = 1– D 1.96 00 (Orl – r1)u^ σο with denoting the cdf of the standard normal distribution. d) Set n = 10, ơ0 = 2, µo = 3. Evaluate numerically the power function for µ = 1,2, 3, 4, 5 and draw its graph on the real axis using R. e) Calculate the density fxa (x) of the third order statistic X(3) under Ho. Hence find numerically P(X(3) < 2). (You could use the integrate function in R ).

Suppose X1, X2, ..., Xn are independent and identically distributed random variables from N(µ, o3) (each with a density f (x; µ) = (z-µ)² 20ó ). Suppose that oo is known. e V2T00 a) Argue that the joint density of X1, X2,..., Xn has monotone likelihood ratio in E X;. i=D1 b) Derive the UMP unbiased size a = 0.05 test p* of Ho : u = µo versus H1 :µ # po. c) Show that the power function of this test is (Or1 – 1)u^ + +-1.96 Eµ4* = 1– D 1.96 00 (Orl – r1)u^ σο with denoting the cdf of the standard normal distribution. d) Set n = 10, ơ0 = 2, µo = 3. Evaluate numerically the power function for µ = 1,2, 3, 4, 5 and draw its graph on the real axis using R. e) Calculate the density fxa (x) of the third order statistic X(3) under Ho. Hence find numerically P(X(3) < 2). (You could use the integrate function in R ).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 31E

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