Part2-Notes-431-2011-F

Part2-Notes-431-2011-F - Review Notes for Loss Models 1 -...

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Review Notes for Loss Models 1 - ACTSC 431/831, FALL 2011 Part 2 – Severity Models 1. Severity models are distributions that are used to model the distribution of the amount of a claim/loss. 2. The distribution function F ( y ) of a random variable Y is called an n -point mixture distribution if F ( y ) = α 1 F X 1 ( y ) + ··· + α n F X n ( y ) for all y, where α j 0, F X j ( y ) is the distribution function of a random variable X j for j = 1 ,...,n , and α 1 + + α n = 1. (a) The survival function of the n -point mixture distribution is S ( y ) = α 1 S X 1 ( y ) + + α n S X n ( y ) for all y. (b) If F X j ( y ) has pdf f X j ( y ) for j = 1 , then the pdf of the n -point mixture distribution is f ( y ) = α 1 f X 1 ( y ) + + α n f X n ( y ) for all y. (c) The k -th moment of the n -point mixture distribution is E ( Y k ) = α 1 E ( X k 1 ) + + α n E ( X k n ) . (d) The moment generating function (mgf) of the n -point mixture distribution is M Y ( t ) = α 1 M X 1 ( t ) + + α n M X n ( t ) , where, the moment generating function (mgf) of a random variable X or its distribution is defined as M X ( t ) = E ( e tX ) wherever this expectation exists for some t > 0. The mgf of X has the following properties: i. M 0 X (0) = E ( X ) and M 00 X (0) = E ( X 2 ). ii. The distribution of a random variable X is determined uniquely by its mo- ment generating function, namely there is a one-to-one relationship between the mgf and its distribution. iii. If random variables X and Y are independent, then M X + Y ( t ) = M X ( t ) M Y ( t ). 1
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3. The tail probability or tail of a random variable X or its distribution F is the probability that X exceeds a positive value t , or Pr { X > t } = 1 - F ( t ) = S ( t ) , t > 0 . (a) Comparison between the tails of two random variables: Suppose that X and Y have distribution functions F X ( x ) and F Y ( x ), respectively. Assume that lim x →∞ Pr { X > x } Pr { Y > x } = lim x →∞ S X ( x ) S Y ( x ) = c. If c = 0, we say that X ( F X ( x )) has a lighter tail than Y ( F Y ( x )); if 0 < c < , we say that X ( F X ( x )) and Y ( F Y ( x )) have proportional tails ; if c = , we say that X ( F X ( x )) has a heavier tail than Y ( F Y ( x )). (b) If X has a heavier tail than Y , then exists a constant x 0 > 0 such that for all x > x 0 , Pr { X > x } > Pr { Y > x } .
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This note was uploaded on 02/08/2012 for the course ACTSC 431 taught by Professor Laundriualt during the Fall '09 term at Waterloo.

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Part2-Notes-431-2011-F - Review Notes for Loss Models 1 -...

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