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April 12th 2006 08:54 PM
Actuary.com - Newbie Poster
mgf (insurance)Question - Australia
im studying actuarial studies in Australia, and i have a problem on moment generating functions (im not sure where this fits in with your curriculum)....can anyone help me?
Claim amounts for a certain insurance portfolio, x1,x2..., follow a distribution with pdf:
f(x;y) = 1/(2y)*e^(-x/y) + 1/y*e^(-2x/y)
calculate the moment generating function of X and show that E(X) = 3/4*y and var(x) = 11/16*y^2
April 13th 2006 01:21 PM
Actuary.com - Level II Poster
I guess the moment generating function turns out to be
Originally Posted by foofighters26
M(t) = 1/(2y*(t-1/y)) + 1/(y*(t-2/y)) [ it is the integral of e^(tx)f(x,y) from 0 < x < infinity; note that t < 1/y otherwise the integral diverges.
<X> = - (3/4)*y [ TAKE the derivative with respect to 't', keeping y fixed and put t = 0] [Note, E[X] is negative, (please check the calculation again)]
<X^2> = (5/4)*y^2 [Take the 2nd Derivative with respect to 't', keeping y fixed and put t = 0]
So, Var(X) = <X^2> - <X>^2 = (11/16)*Y^2
Does that help ?
If you need to see all the steps, let me know ( but it is kinda hard to write down those mathematical symbols though ).
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