September 2012
Beginner to intermediate
536 pages
14h 40m
English
book
Loss Models: From Data to Decisions, 4th Edition
by Stuart A. Klugman, Harry H. Panjer, Gordon E. Willmot
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E.3 Undiscretization of a discretized distribution
Assume we have g0 = Pr(S = 0), the true probability that the random variable is zero. Let pj = Pr(S* = jh), where S* is a discretized distribution and h is the span. The following are approximations for the cdf and limited expected value of S, the true distribution that was discretized as S*. They are all based on the assumption that S has a uniform distribution over the interval from (j − 
)h to (j + 
)h for integral j. The first interval is from 0 to h/2, and the probability p0 − g0 is assumed to be uniformly distributed over it. Let S** be the random variable with this approximate mixed distribution. (It is continuous, except for discrete probability g0 at zero.) The approximate distribution function can be found by interpolation as follows. First, let
Then, for x in the interval (j − 1/2)h to (j + 1/2)h,
Because the first interval is only half as wide, the formula for 0 ≤ x ≤ h/2 is
It is also possible to express these formulas in terms ...
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