02_Exam_Solution

02_Exam_Solution - Exam 2 - Solution ORIE 3500/5500:...

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Unformatted text preview: Exam 2 - Solution ORIE 3500/5500: Engineering Probability and Statistics II Question 1 (i) (a) Conditional PDF of X given Y f X | Y ( x | y ) is a function with f X | Y ( x | y ) ≥ 0 and R ∞-∞ f X | Y ( x | y ) dx = 1 such that P ( X ≤ a | Y = y ) = R a-∞ f X | Y ( x | y ) dx . Also f X | Y ( x | y ) = f X,Y ( x,y ) f Y ( y ) , where f X,Y is the joint PDF of X and Y and f Y is the marginal PDF of Y . (b) X and Y will be independent if f X,Y ( x,y ) = f X ( x ) f Y ( y ) , where f X,Y is the joint PDF of X and Y and f X ,f Y are the marginal PDF of X and Y . (c) Moment-generating function (transform) of a random variable X is M X ( s ) = E e s · X . (d) Moment-generating function (transform) of two random variables X and Y is a func- tion M X,Y ( s 1 ,s 2 ) such that M X,Y ( s 1 ,s 2 ) = E e s 1 · X + s 2 · Y . (ii) (a) M Y ( s ) = E e s · Y = E h e s · ( a + bX ) i = E h e sa e s · bX i = e sa M X ( bs ) ....
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This homework help was uploaded on 02/20/2009 for the course ORIE 3500 taught by Professor Weber during the Fall '08 term at Cornell.

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02_Exam_Solution - Exam 2 - Solution ORIE 3500/5500:...

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