Dr. Hackney STA Solutions pg 49

Dr. Hackney STA Solutions pg 49 - T = R 2 and θ is f T,θ...

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Second Edition 4-5 4.20 a. This transformation is not one-to-one because you cannot determine the sign of X 2 from Y 1 and Y 2 . So partition the support of ( X 1 ,X 2 ) into A 0 = {-∞ < x 1 < ,x 2 = 0 } , A 1 = {-∞ < x 1 < ,x 2 > 0 } and A 2 = {-∞ < x 1 < ,x 2 < 0 } . The support of ( Y 1 ,Y 2 ) is B = { 0 < y 1 < , - 1 < y 2 < 1 } . The inverse transformation from B to A 1 is x 1 = y 2 y 1 and x 2 = p y 1 - y 1 y 2 2 with Jacobian J 1 = ± ± ± ± ± ± 1 2 y 2 y 1 y 1 1 2 1 - y 2 2 y 1 y 2 y 1 1 - y 2 2 ± ± ± ± ± ± = 1 2 p 1 - y 2 2 . The inverse transformation from B to A 2 is x 1 = y 2 y 1 and x 2 = - p y 1 - y 1 y 2 2 with J 2 = - J 1 . From (4.3.6), f Y 1 ,Y 2 ( y 1 ,y 2 ) is the sum of two terms, both of which are the same in this case. Then f Y 1 ,Y 2 ( y 1 ,y 2 ) = 2 " 1 2 πσ 2 e - y 1 / (2 σ 2 ) 1 2 p 1 - y 2 2 # = 1 2 πσ 2 e - y 1 / (2 σ 2 ) 1 p 1 - y 2 2 , 0 < y 1 < , - 1 < y 2 < 1 . b. We see in the above expression that the joint pdf factors into a function of y 1 and a function of y 2 . So Y 1 and Y 2 are independent. Y 1 is the square of the distance from ( X 1 ,X 2 ) to the origin. Y 2 is the cosine of the angle between the positive x 1 -axis and the line from ( X 1 ,X 2 ) to the origin. So independence says the distance from the origin is independent of the orientation (as measured by the angle). 4.21 Since R and θ are independent, the joint pdf of
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Unformatted text preview: T = R 2 and θ is f T,θ ( t,θ ) = 1 4 π e-t/ 2 , < t < ∞ , < θ < 2 π. Make the transformation x = √ t cos θ , y = √ t sin θ . Then t = x 2 + y 2 , θ = tan-1 ( y/x ), and J = ± ± ± ± 2 x 2 y-y x 2 + y 2-x x 2 + y 2 ± ± ± ± = 2 . Therefore f X,Y ( x,y ) = 2 4 π e-1 2 ( x 2 + y 2 ) , < x 2 + y 2 < ∞ , < tan-1 y/x < 2 π. Thus, f X,Y ( x,y ) = 1 2 π e-1 2 ( x 2 + y 2 ) ,-∞ < x,y < ∞ . So X and Y are independent standard normals. 4.23 a. Let y = v , x = u/y = u/v then J = ± ± ± ± ∂x ∂u ∂x ∂v ∂y ∂u ∂y ∂v ± ± ± ± = ± ± ± ± 1 v-u v 2 1 ± ± ± ± = 1 v . f U,V ( u,v ) = Γ( α + β ) Γ( α )Γ( β ) Γ( α + β + γ ) Γ( α + β )Γ( γ ) ² u v ³ α-1 ² 1-u v ³ β-1 v α + β-1 (1-v ) γ-1 1 v , < u < v < 1 ....
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.

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