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Probability and Statistics Quiz Two Solution 2 2 1 2 2 , 1. : cos sin tan and are two i.i.d. Gaussian random variables with zero mean and unit variance. 1 The joint PDF ( , ) ( ) ( ) exp{ }. 2 2 By Jacobian transformation, we c X Y X Y Sol R X Y X R Y Y R X X Y x y f x y f x f y π - = + = Φ = Φ Φ = + = = - 2 , , 2 , , 0 , an obtain the joint PDF as following: 1 ( , ) ( , ) exp{ } , 0. 2 2 (1) The PDF of : ( ) ( , ) exp{ } ( ). 2 (2) The PDF of : 1 ( ) ( , ) , [ , ). 2 1 ( , ) exp{ 2 R X Y R R R R dxdy r f r f x y r r drd R r f r f r d r u r f f r d f r r π π φ φ π φ φ φ φ φ φ π π π φ π Φ Φ - Φ Φ Φ = = - = = - Φ = = - = - 2 } ( ) ( ) ( ). 2 and are independent. R r u r f r f R φ Φ = Φ , , 0 0 2. : and are two i.i.d. exponential random variables. The joint PDF ( , ) ( ) ( ) exp{ ( )} ( ) ( ). Let

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