solutions_ch5_exercises

solutions_ch5_exercises - Chapter 5 Properties of a Random...

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Unformatted text preview: Chapter 5 Properties of a Random Sample 5.1 Let X = # color blind people in a sample of size n . Then X binomial( n,p ), where p = . 01. The probability that a sample contains a color blind person is P ( X > 0) = 1- P ( X = 0), where P ( X = 0) = ( n ) ( . 01) ( . 99) n = . 99 n . Thus, P ( X > 0) = 1- . 99 n > . 95 n > log( . 05) / log( . 99) 299 . 5.3 Note that Y i Bernoulli with p i = P ( X i ) = 1- F ( ) for each i . Since the Y i s are iid Bernoulli, n i =1 Y i binomial( n,p = 1- F ( )). 5.5 Let Y = X 1 + + X n . Then X = (1 /n ) Y , a scale transformation. Therefore the pdf of X is f X ( x ) = 1 1 /n f Y parenleftBig x 1 /n parenrightBig = nf Y ( nx ). 5.6 a. For Z = X- Y , set W = X . Then Y = W- Z , X = W , and | J | = vextendsingle vextendsingle vextendsingle vextendsingle 1- 1 1 vextendsingle vextendsingle vextendsingle vextendsingle = 1 . Then f Z,W ( z,w ) = f X ( w ) f Y ( w- z ) 1, thus f Z ( z ) = integraltext - f X ( w ) f Y ( w- z ) dw . b. For Z = XY , set W = X . Then Y = Z/W and | J | = vextendsingle vextendsingle vextendsingle vextendsingle 1 1 /w- z/w 2 vextendsingle vextendsingle vextendsingle vextendsingle =- 1 /w . Then f Z,W ( z,w ) = f X ( w ) f Y ( z/w ) |- 1 /w | , thus f Z ( z ) = integraltext - |- 1 /w | f X ( w ) f Y ( z/w ) dw . c. For Z = X/Y , set W = X . Then Y=W/Z and | J | = vextendsingle vextendsingle vextendsingle vextendsingle 1- w/z 2 1 /z vextendsingle vextendsingle vextendsingle vextendsingle = w/z 2 . Then f Z,W ( z,w ) = f X ( w ) f Y ( w/z ) | w/z 2 | , thus f Z ( z ) = integraltext - | w/z 2 | f X ( w ) f Y ( w/z ) dw . 5.7 It is, perhaps, easiest to recover the constants by doing the integrations. We have integraldisplay - B 1+ ( ) 2 d = B, integraldisplay - D 1+ ( - z ) 2 d = D and integraldisplay - bracketleftBigg A 1+ ( ) 2- C 1+ ( - z ) 2 bracketrightBigg d = integraldisplay - bracketleftBigg A 1+ ( ) 2- C ( - z ) 1+ ( - z ) 2 bracketrightBigg d- Cz integraldisplay - 1 1+ ( - z ) 2 d = A 2 2 log...
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solutions_ch5_exercises - Chapter 5 Properties of a Random...

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