1 Pad the input by adding a zero outcome X = 0 with probability 0.

      2 Group by Xj and sum the corresponding pj.

      3 Sort events by outcome Xj into ascending order. Relabel events X0<X1<⋯<Xm and probabilities p0,…,pm.

      4 Decumulate probabilities to compute the survival function Sj:=S(Xj) using S0=1−p0 and Sj=Sj−1−pj, j > 0.

      5 Difference the outcomes to compute ΔXj=Xj+1−Xj, j=0,…,m−1.

      6 Outcome-probability sum-product: (3.10)

      7 Survival function sum-product: (3.11)

      1 Step (1) treats 0 as special because the second integral in Eq. 3.9 starts at X = 0. The case where the smallest outcome is > 0 is illustrated in Figure 3.12. Now S(x)=1 for 0≤x<X1 and ∫0∞S(x)dx includes the shaded dashed rectangle of area X1. Step (1) allows us to systematically deal with any discrete data. It adds a new outcome row only when the smallest observation is > 0.Figure 3.12 Accounting for the effect of adding 1 to each outcome.

      2 After Step (3), the Xj are distinct, they are in ascending order, X0=0, and pj=P(X=Xj).

      3 In Step (4), Sm=P(X>Xm)=0 since Xm is the maximum value of X.

      4 The forward difference ΔX computed in Step (5) replaces dx in various formulas. Since it is a forward difference ΔXm is undefined. It is also unneeded.

      5 Step (6) computes the first integral in Eq. 3.9. It is a sum because X is discrete and has a probability mass function, like a Poisson, rather than a density, like a normal—explaining why we use the notation dF(x), not f(x)dx. The sum starts at i = 1 because X0=0. Notice that P(X=Xj)=Sj−1−Sj is the negative backward difference of S.

      6 Step (7) computes the second integral in Eq. 3.9. The sum starts at i = 1, corresponding to the first vertical bar in Panel (b) that extends from X0=0 to X1 and has height S(X0).

      7 Note the index shift between Eq. 3.10, 3.11.

      8 Both Eqs. 3.10 and 3.11 are exact evaluations. The approximation occurs when the underlying distribution being modeled is replaced with the discrete sample given by X.