(a) Let X be a random variable of the continuous type with probability density function f(x). which is positive provided 0
(a) Let X be a random variable of the continuous type with probability density function f(x). which is positive provided 0
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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![(a) Let X be a random variable of the continuous type with probability density function f(x),
which is positive provided 0<x<a<o, and is equal to zero elsewhere. Show that
E(X)= [1-F(x)]dx, where F(x) is the cumulative distribution function of X.
(b) Let 0<p<1. A (100p)h percentile (quantile of order p) of the distribution of a continuous
random variable X is a value 5, such that P(X s 5.) = p. Consider a random variable that
has the probability density function f(x) 4x' for 0<x<1, and zero otherwise.
(i) Find 5, in terms of the probability p.
(ii) Find the 20th percentile of the distribution.
(iii) A CMU faculty's salary is at the 20th percentile. Explain the meaning of this percentile.
(iv) Using your result in (i) or otherwise, what percentile is x 0.99?
(c) Let Y, <Y, <Y, <Y, <Y, be the order statistics of a random sample of size n= 5 from a
uniform distribution with probability density function f(x;0)=, 0<x<0, zero elsewhere,
where 0 > 0. The hypothesis H, :0=1 is rejected and H,:0>1 is accepted if the observed
value Y, 2 c.
(i) Find the constant c so that the significance level is a = 0.05.
(ii) Determine the power function of the test. Find the power when 0= 2.
(iii) Using your result in (ii), what is the power of the test when H, is true.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F51b99ea9-2545-4b2e-baa7-dca4acbd53ba%2F821e2167-5fd2-4461-9141-f1616f55b693%2Fop3hhfi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Let X be a random variable of the continuous type with probability density function f(x),
which is positive provided 0<x<a<o, and is equal to zero elsewhere. Show that
E(X)= [1-F(x)]dx, where F(x) is the cumulative distribution function of X.
(b) Let 0<p<1. A (100p)h percentile (quantile of order p) of the distribution of a continuous
random variable X is a value 5, such that P(X s 5.) = p. Consider a random variable that
has the probability density function f(x) 4x' for 0<x<1, and zero otherwise.
(i) Find 5, in terms of the probability p.
(ii) Find the 20th percentile of the distribution.
(iii) A CMU faculty's salary is at the 20th percentile. Explain the meaning of this percentile.
(iv) Using your result in (i) or otherwise, what percentile is x 0.99?
(c) Let Y, <Y, <Y, <Y, <Y, be the order statistics of a random sample of size n= 5 from a
uniform distribution with probability density function f(x;0)=, 0<x<0, zero elsewhere,
where 0 > 0. The hypothesis H, :0=1 is rejected and H,:0>1 is accepted if the observed
value Y, 2 c.
(i) Find the constant c so that the significance level is a = 0.05.
(ii) Determine the power function of the test. Find the power when 0= 2.
(iii) Using your result in (ii), what is the power of the test when H, is true.
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