I was reading the article about the Schuette–Nesbitt formula, which is described as "a generalization of the inclusion–exclusion principle", which has both a combinatorial and probabilistic versions. Another website gave a proof for dependent events (pdf download), and found a third that compares it to Waring's Theorem (pdf)

However, I am still confused. I tried finding a clear worked-out example using discrete probabilities (for simplicity) that the steps are clear from one line to the next - to help in overall understanding of the formula.

Is there a good reference, or an answer that can give a short worked-out example?


I have found an example in following book and my answer is a modified version of the Sec 8.4,8.6 of the book in order to make it concise and clear.

Gerber, Hans U. "Life insurance." Life Insurance Mathematics. Springer Berlin Heidelberg, 1990.

$B_1,\cdots B_n$ are arbitrary events. $N$ is a random variable ranging over $\{0, 1, ... , m\}$. For arbitrary real coefficients $c_1,\cdots c_m$, the Schuette–Nesbitt formula is the following operator identity between shifting operator $E:c_n\mapsto c_{n+1}$ and difference operator $\Delta:c_n\mapsto c_{n+1}-c_{n}$. By definition they are related via $E=id+\Delta$, the SN formula is $$\sum_{n=0}^{m}c_n\cdot Pr(N=n)=\sum_{k=0}^{m}[\Delta^{k}c_0]S_k$$ where $S_k=\sum_{j_1,\cdots j_k}Pr(B_{j1}\cap\cdots \cap B_{jk})$ is the symmetric sum among these $n$ events and $S_0=1$. Note that $[\Delta^{k}c_0]$ means difference operator acting on $c_0$. For example, $[\Delta^{2}c_0]=\Delta^{1}(c_1-c_0)=\Delta^{1}(c_1)-\Delta^{1}(c_0)=(c_2-c_1)-(c_1-c_0)=c_2-2c_1+c_0$. Both operators are linear and hence they have representations in terms of matrix, therefore they can be extended to polynomial rings and modules (since these two objects have "basis",loosely speaking.) $$E=\left(\begin{array}{ccccc} 0 & 0 & 0 & \cdots\\ 1 & 0 & 0 & \cdots\\ 0 & 1 & 0 & \cdots\\ 0 & 0 & 1 & \cdots \end{array}\right) $$ $$\Delta=\left(\begin{array}{ccccc} -1 & 0 & 0 & \cdots\\ 1 & -1 & 0 & \cdots\\ 0 & 1 & -1 & \cdots\\ 0 & 0 & 1 & \cdots \end{array}\right)$$

The proof makes use of the indicator trick and expansion of the operator polynomial $\prod_{j=1}^{m}(1+I_{B_j}\Delta)$ and the fact that $I_A\cdot I_B=I_{A\cap B}$ and $\Delta$ commutes with indicators, I will refer you to Gerber's book.

If we choose $c_0=1$ and all other $c_1=c_2=\cdots=c_n=1$, then SN formula becomes the inclusion-exclusion principle as below: $$\sum_{n=1}^{m} Pr(N=n)=\sum_{k=0}^{m}\Delta^{k}c_0S_k=c_0 S_0+(c_1-c_0)S_1+(c_2-2c_1+c_0)S_2+\cdots =S_1-S_2+S_3+\cdots+(-1)^{n}S_n=[Pr(B_1)+\cdots+Pr(B_n)]-[Pr(B_1\cap B_2)+\cdots+Pr(B_{n-1}\cap B_{n})]+\cdots+(-1)^n\cdot Pr(S_1\cap\cdots \cap S_n)$$

Waring's Theorem gives the probability that exactly $r$ out of the $n$ events $B_1,\cdots B_n$ occur. Thus it can be derived by specifying $c_r=1$ and all other $c$'s=0. The SN formula becomes $$ Pr(N=r)=\sum_{k=0}^{m}[\Delta^{k}c_0]S_k=\sum_{k=r}^{m}[\Delta^{k}c_0]S_k$$ because any term $[\Delta^{k}c_0]=0$ when $k<r$, a change of variable $t=k-r$ will yield Waring's formula.

There is an envelope assignment example in Gerber's book you can have a look into, but my suggestion is to understand it in terms of operator algebra instead of probability.

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