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# 02_01 - e t 2 0.20 e t 1 t 2 0.25 e t 1 2 t 2 0.30 e 2 t 1...

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STAT 410 Examples for 02/01/2008 Spring 2008 Moment-generating function M X Y ( t 1 , t 2 ) = E ( e t 1 X + t 2 Y ) , if it exists for | t 1 | < h 1 , | t 2 | < h 2 . M X Y ( t 1 , 0 ) = M X ( t 1 ) , M X Y ( 0, t 2 ) = M Y ( t 2 ) . Example 1 : Consider the following joint probability distribution p ( x , y ) of two random variables X and Y: x \ y 0 1 2 1 0.15 0.15 0 2 0.15 0.35 0.20 Find the moment-generating function M X Y ( t 1 , t 2 ) . M ( t 1 , t 2 ) = 0.15 e t 1 + 0.15 e 2 t 1 + 0.15 e t 1 + t 2 + 0.35 e 2 t 1 + t 2 + 0.20 e 2 t 1 + 2 t 2 . Example 2 : Consider two random variables X and Y with the moment-generating function M ( t 1 , t 2 ) = 0.10 e 2 t 1 + 0.15 e t 2 + 0.20 e t 1

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Unformatted text preview: e t 2 + 0.20 e t 1 + t 2 + 0.25 e t 1 + 2 t 2 + 0.30 e 2 t 1 + 2 t 2 . Find the joint probability density function p ( x , y ). x \ y 0 1 2 0 0 0.15 0 1 0 0.20 0.25 2 0.10 0 0.30 Example 3 : Let the joint probability density function for ( X , Y ) be ( ) & ± ² ≤ + ≤ ≤ ≤ ≤ = otherwise 1 , 1 , 1 24 , y x y x y x y x f Find the moment-generating function M ( t 1 , t 2 ). M ( t 1 , t 2 ) = ³ ³ ´ ´ µ ¶ · · ¸ ¹-+ ⋅ 1 1 24 2 1 dx dy y x e x y t x t = …...
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