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Unformatted text preview: STAT 410 Summer 2009 Homework #2 (due Wednesday, June 24, by 4:00 p.m.) 1. The time, T, that a manufacturing system is out of operation has cumulative distribution function F ( t ) =  otherwise 2 if 2 1 2 t t The resulting cost to the company is Y = T 2 . Determine the density function of Y. t > 2 Y = T 2 y > 4 F Y ( y ) = P ( Y y ) = P ( T 2 y ) = P ( T y ) = y 4 1 , y > 4. f Y ( y ) = F Y ' ( y ) = otherwise 4 if 4 2 y y OR f T ( t ) = F T ' ( t ) = otherwise 2 if 8 3 t t g ( t ) = t 2 g 1 ( y ) = y d t / d y = y 2 1 = 2 1 y 1 / 2 f Y ( y ) = f T ( g 1 ( y ) ) y t d d = ( 8 y 3 / 2 ) ( 2 1 y 1 / 2 ) = 4 y 2 , y > 4 2. Grades on the last STAT 410 exam were not very good. Graphed, their distribution had a shape similar to the p.d.f. f X ( x ) = 000 , 5 1 ( 100 x ), 0 x 100, zero elsewhere. a) What was the class average, E ( X ) ? E ( X ) = ( 29  100 100 000 , 5 1 dx x x = 3 33.33 . As a way of curving the results, the professor announces that he will replace each persons grade, X, with a new grade, Y = g ( X ), where g ( X ) = 10 X . b) Find the p.d.f. that describes the new grades, Y. y = 10 x x = 100 2 y dy dx = 50 y f Y ( y ) = 50 100 100 000 , 5 1 2 y y  = ( 29 2 000 , 10 000 , 000 , 25 1 y y , 0 y 100. c) Has the professors strategy been successful in raising the class average above 60? What is the new class average, E ( Y ) ? E ( Y ) = ( 29  100 100 000 , 5 1 10 dx x x = 3 53.33 . OR E ( Y ) = ( 29  100 2 000 , 10 000 , 000 , 25 1 dy y y y = 3 53.33 . 3. Suppose that X follows a uniform distribution on the interval [ / 2 , / 2 ] . Find the c.d.f. and the p.d.f. of Y = tan X. f X ( x ) = < < o.w. 2 2 1 x F X ( x ) = <  + < 2 1 2 2 2 1 2 x x x x F Y ( y ) = P ( Y y ) = P ( tan X y ) = P ( X arctan ( y ) ) = ( 29 2 1 1 arctan + y , < y < . f Y ( y ) = ( 29 2 1 1 y + , < y < . ( Standard ) Cauchy distribution. OR g ( x ) = tan x g 1 ( y ) = arctan ( y ) d x / d y = 2 1 1 y + f Y ( y ) = f X ( g 1 ( y ) ) y x d d = + 2 1 1 1 y = ( 29 2 1 1 y + , < y < . F Y ( y ) = ( 29 + y du u 2 1 1 = ( 29 2 1 1 arctan + y , < y < . 4. 1.7.21 If the p.d.f. of X is f ( x ) = 2 x e x 2 , 0 < x < , zero elsewhere, determine the p.d.f. of Y = X 2 . f X ( x ) = < < o.w....
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This note was uploaded on 10/29/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
 STEPANOV
 Statistics

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