(a) Let X be a random variable of the continuous type with probability density function f(x), which is positive provided 0 0. The hypothesis H, :0=1 is rejected and H,:0>1 is accepted if the observed...

(a) Let X be a random variable of the continuous type with probability density function f(x),<br>which is positive provided 0<x<a<o, and is equal to zero elsewhere. Show that<br>E(X)= [1-F(x)]dx, where F(x) is the cumulative distribution function of X.<br>(b) Let 0<p<1. A (100p)h percentile (quantile of order p) of the distribution of a continuous<br>random variable X is a value 5, such that P(X s 5.) = p. Consider a random variable that<br>has the probability density function f(x) 4x' for 0<x<1, and zero otherwise.<br>(i) Find 5, in terms of the probability p.<br>(ii) Find the 20th percentile of the distribution.<br>(iii) A CMU faculty's salary is at the 20th percentile. Explain the meaning of this percentile.<br>(iv) Using your result in (i) or otherwise, what percentile is x 0.99?<br>(c) Let Y, <Y, <Y, <Y, <Y, be the order statistics of a random sample of size n= 5 from a<br>uniform distribution with probability density function f(x;0)=, 0<x<0, zero elsewhere,<br>where 0 > 0. The hypothesis H, :0=1 is rejected and H,:0>1 is accepted if the observed<br>value Y, 2 c.<br>(i) Find the constant c so that the significance level is a = 0.05.<br>(ii) Determine the power function of the test. Find the power when 0= 2.<br>(iii) Using your result in (ii), what is the power of the test when H, is true.<br>

Extracted text: (a) Let X be a random variable of the continuous type with probability density function f(x), which is positive provided 01 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.

Jun 05, 2022

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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 0. The hypothesis H, :0=1 is rejected and H,:0>1 is accepted if the observed...

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