Loss Data Analysis

where

$f_S\left(x\right)={\displaystyle \sum _{k=0}^{\infty }{p}_k{f}_X^{*k}\left(x\right)},$

where again we use fX(x) instead of P(X = x).

To perform the convolutions, we think of the middle points of the bins in Table 5.10 as the observed losses. Their probabilities obtained from that table are listed below in Table 5.12.

We chose the maximum number of losses to be rather small so that performing the convolutions as in Example 1 is easy. Doing that produces Table 5.13 shown below, in which we compute P(S = x) ≡ fS(x) from x = 0 up to x = 50.

Tab. 5.13: Distribution of S.

Summing up the entries along the even columns of Table 5.13 we obtain F S(50) = ${\displaystyle {\sum }_{s=0}^{50}{f}_S\left(S\right)}=\mathrm{0.473539.}$P(S > 50) = 1 − FS(50) = 0.526461, ...