09_19 - STAT 410 Examples for 09/19/2011 Fall 2011 1. When...

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STAT 410 Examples for 09/19/2011 Fall 2011 1. When a person applies for citizenship in Neverland, first he/she must wait X years for an interview, and then Y more years for the oath ceremony. Thus the total wait is W = X + Y years. Suppose that X and Y are independent, the p.d.f. of X is f X ( x ) = 2 / x 3 , x > 1, zero otherwise, and Y has a Uniform distribution on interval ( 0, 1 ). Find the p.d.f. of W, f W ( w ) = f X + Y ( w ). Hint: Consider two cases: 1 < w < 2 and w > 2. V = Y X = W – V W = X + Y Y = V J = 0 1 1 1 - = – 1 f V, W ( v , w ) = f X, Y ( w v , v ) × 1 = ( ) 3 2 v w - . x > 1 w v > 1 w > v + 1 0 < y < 1 0 < v < 1 1 < w < 2 f W ( w ) = ( ) - - 1 0 3 2 w dv v w = ( ) 0 1 2 1 - - w v w = 2 1 1 w - . w > 2 f W ( w ) = ( ) - 1 0 3 2 dv v w = ( ) 0 1 2 1 - v w = ( ) 2 2 1 1 1 w w - - .
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OR V = X X = V W = X + Y Y = W – V J = 1 1 0 1 - = 1 f V, W ( v , w ) = f X, Y ( v , w v ) × 1
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09_19 - STAT 410 Examples for 09/19/2011 Fall 2011 1. When...

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