370mgfproblems

370mgfproblems - Math 370, Actuarial Problemsolving...

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Math 370, Actuarial Problemsolving Moment-generating functions Moment-generating functions Definitions and properties General definition of an mgf: M ( t ) = M X ( t ) = E( e tX ). Mgf of discrete r.v.’s: M ( t ) = e tx f ( x ), where f ( x ) is the p.m.f. of X and the sum is taken over all values x of X . Mgf of continuous r.v.’s: M ( t ) = R * * e tx f ( x ) dx , where f ( x ) is the p.d.f. of X , and the range of integration is the range on which f ( x ) is defined. Expectation, variance, and moments via mgf’s: M (0) = 1, M 0 (0) = E( X ), M 00 (0) = E( X 2 ), M 000 (0) = E( X 3 ), etc.; Var( X ) = M 00 (0) - M 0 (0) 2 . Mgf of a multiple of a r.v.: If X has mgf M X ( t ), and Y = cX with c a constant, then the mgf of Y is M Y ( t ) = E ( e tY ) = E ( e tcX ) = M X ( tc ). Mgf of a sum of independent r.v.’s X and Y : If X and Y are independent, then X + Y has mgf M X + Y ( t ) = M X ( t ) M Y ( t ). (An analogous formula holds for sums of more than two independent
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370mgfproblems - Math 370, Actuarial Problemsolving...

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