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 ).

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

1. Suppose X1, X2, ..., Xn are independent and identically distributed random variables from
1
N(µ, o3) (each with a density f(x; µ) =
206 ). Suppose that oo is known.
V2n00
a) 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 is
Vn(µ – H0)
Vn(u – Ho)
Euf* = 1 – ¢
1.96
-1.96
σο
σο
with 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, 5
and 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 find
numerically P(X (3) < 2). (You could use the integrate function in R ).

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