HW09_10soln - -sin-1(y Case when Y>0 x P(Y

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y x Case when Y>0 P(Y<y)=1-P(a<X<b) x x y y a a b b sin -1 (y) is negative here π -sin -1 (y) x x y y 2 π +sin -1 (y) sin -1 (y) π -sin -1 (y) sin -1 (y) is positive here IE 111 Homework #9 Question 1 A random variable X has density function f X (x) = 2x for 0<x<1 a) Let Y = X 2 . Find the density function of random variable Y The function is increasing over the range 0<x<1, thus Range of Y is 0≤y≤1 OR the easy way
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b) Let Y = (X-0.5) 2 . Find the density function of random variable Y Now the function is neither increasing nor decreasing over 0<x<1. Thus we must do it the hard way. Range of Y is 0≤y≤0.25 c) Let Y = (X-0.25) 2 . Find the density function of random variable Y Again the function is neither increasing nor decreasing over 0<x<1. Thus we must do it the hard way. Range of Y is 0 ≤ y ≤ 0.75 2 = 0.5625
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The table below shows the upper and lower limits of the inequality above for various values of y. y Probability 0 0.25 0.25 0 0.04=0.2 2 0.45 0.05 F X (0.45)- F X (0.05) 0.0625=0.25 2 0.5 0 F X (0.5)- F X (0) 0.25=0.5 2 0.75 -0.25 F X (0.5)- F X (0) (can’t plug -0.25 into F X ()) 0.5625=0.75 2 1 -0.5
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This note was uploaded on 09/06/2010 for the course IE 111 taught by Professor Storer during the Spring '07 term at Lehigh University .

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HW09_10soln - -sin-1(y Case when Y>0 x P(Y

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