elemprob-page25 - Proposition 12.2. If m X ( t ) = m Y ( t...

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Unformatted text preview: Proposition 12.2. If m X ( t ) = m Y ( t ) < for all t in an interval, then X and Y have the same distribution. We will not prove this, but this is essentially the uniqueness of the Laplace transform. Note E e tX = R e tx f X ( x ) dx . If f X ( x ) = 0 for x < 0, this is R e tx f X ( x ) dx = L f X (- t ), where L f X is the Laplace transform of f X . We can use this to verify some of the properties of sums we proved before. For example, if X is a N ( a,b 2 ) and Y is a N ( c,d 2 ) and X and Y are independent, then m X + Y ( t ) = e at + b 2 t 2 / 2 e ct + d 2 t 2 / 2 = e ( a + c ) t +( b 2 + d 2 ) t 2 / 2 . Proposition 12.2 then implies that X + Y is a N ( a + c,b 2 + d 2 ). Similarly, if X and Y are independent Poisson random variables with parameters a and b , resp., then m X + Y ( t ) = m X ( t ) m Y ( t ) = e a ( e t- 1) e b ( e t- 1) = e ( a + b )( e t- 1) , which is the moment generating function of a Poisson with parameter a + b ....
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