Loss Models: From Data to Decisions, 4th Edition

E.3 Undiscretization of a discretized distribution

Assume we have g₀ = Pr(S = 0), the true probability that the random variable is zero. Let p_j = 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 ...