s410_hw3ans - STAT 410 HW #3 Answers Due Wednesday, October...

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STAT 410 HW #3 Answers Due Wednesday, October 15, 2003 You can turn it in to my office, 116B IH, or mailbox in 101 IH. 1 . The Gamma Distribution has two parameters, the shape p > 0 and the scale λ > 0, and is denoted Gamma ( p, λ ). Its space is (0 , ), and the pdf is f ( y ; p, λ ) = c y p - 1 e - λy . (a) Show that c = λ p Γ( p ) . Answer: 1 c = Z 0 y p - 1 e - λy dy (set u = λy ) = Z 0 ( u/λ ) p - 1 e - u du/λ = 1 λ p Z 0 u p - 1 e - u du = 1 λ p Γ( p ) . (b) χ 2 ν and Exponential ( λ ) are special cases of the Gamma. What are the corresponding ( p, λ )’s? Answer: χ 2 ν = Gamma ( ν 2 , 1 2 ) and Exponential ( λ ) = Gamma (1 , λ ). (c) If Y Gamma ( p, λ ), then what is the distribution of aY for positive constant a ? Answer: Let X = aY , so that y = x/a and the Jacobian is 1 /a . The the pdf of X is f X ( x ) = f Y ( x/a ) /a = λ p Γ( p ) ( y/a ) p - 1 e - λy/a /a = ( λ/a ) p Γ( p ) y p - 1 e - ( λ/a ) y , which is Gamma ( p, λ/a ). 2 . Suppose X 1 and X 2 are independent, with X 1 Gamma ( p 1 , λ ) and X 2 Gamma ( p 2 , λ ). Let Y 1 = X 1 + X 2 and Y 2 = X 1 X 1 + X 2 . (a) What is the space of ( Y 1 , Y 2 )? 1
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Answer: Y = (0 , ) × (0 , 1). (b) Find g - 1 ( y 1 , y 2 ) and the Jacobian. Answer: x 1 = y 1 y 2 and x 2 = y 1 - x 1 = y 1 (1 - y 2 ), so g - 1 ( y 1 , y 2 ) = ( y 1 y 2 , y 1 (1 - y 2 )). The Jacobian is then ± ± ± ± ± ± ∂y 1 y 2 ∂y 1 ∂y 1 (1 - y 2 ) ∂y 1 ∂y 1 y 2 ∂y 2 ∂y 1 (1 - y 2 ) ∂y 2 ± ± ± ± ± ± = ± ± ± ± ± y 2 1 - y 2 y 1 - y 1 ± ± ± ± ± = - y 2 y 1 - y 1 (1 - y 2 ) = - y 1 . (c) Find the pdf of ( Y 1 , Y 2 ). Answer: f X ( x 1 , x 2 ) = λ p 1 Γ( p 1 ) λ p 2 Γ( p 2 ) x p 1 - 1 1 x p 2 - 1 2 e - λ ( x 1 + x 2 ) , so f Y ( y 1 , y 2 ) = f X ( y 1 y 2 , y 1 (1 - y 2 )) | - y 1 | = λ p 1 Γ( p 1 ) λ p 2 Γ( p 2 ) ( y 1 y 2 ) p 1 - 1 ( y 1 (1 - y 2 )) p 2 - 1 e - λy 1 y 1 = λ p 1 + p 2 Γ( p 1 )Γ( p 2 ) y p 1 + p 2 - 1 1 e - λy 1 y p 1 - 1 2 (1 - y 2 ) p 2 - 1 (d) Are Y 1 and Y 2 independent? What are their distributions? (They should be Gamma and Beta, respectively.) How does Y 2 depend on λ ? Answer: Yes, they are independent, because their joint pdf factors into a function of just
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This note was uploaded on 09/08/2011 for the course STAT 410 taught by Professor Monrad during the Spring '08 term at University of Illinois, Urbana Champaign.

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s410_hw3ans - STAT 410 HW #3 Answers Due Wednesday, October...

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