Schuette–Nesbitt formula

In mathematics, the Schuette–Nesbitt formula is a generalization of the inclusion–exclusion principle. It is named after Donald R. Schuette and Cecil J. Nesbitt.

The probabilistic version of the Schuette–Nesbitt formula has practical applications in actuarial science, where it is used to calculate the net single premium for life annuities and life insurances based on the general symmetric status.

Combinatorial versions

Consider a set Template:Math and subsets Template:Math. Let

denote the number of subsets to which Template:Math belongs, where we use the indicator functions of the sets Template:Math. Furthermore, for each Template:Math, let

denote the number of intersections of exactly Template:Mvar sets out of Template:Math, to which Template:Mvar belongs, where the intersection over the empty index set is defined as Template:Math, hence Template:Math. Let Template:Mvar denote a vector space over a field Template:Mvar such as the real or complex numbers (or more generally a module over a ring Template:Mvar with multiplicative identity). Then, for every choice of Template:Math,

where Template:Math denotes the indicator function of the set of all Template:Math with Template:Math, and $(\genfrac{}{}{0ex}{}{k}{l})$ is a binomial coefficient. Equality (Template:EquationNote) says that the two Template:Mvar-valued functions defined on Template:Math are the same.

Proof of (Template:EquationNote)

We prove that (Template:EquationNote) holds pointwise. Take Template:Math and define Template:Math. Then the left-hand side of (Template:EquationNote) equals Template:Math. Let Template:Mvar denote the set of all those indices Template:Math such that Template:Math, hence Template:Mvar contains exactly Template:Mvar indices. Given Template:Math with Template:Mvar elements, then Template:Mvar belongs to the intersection Template:Math if and only if Template:Mvar is a subset of Template:Mvar. By the combinatorial interpretation of the binomial coefficient, there are Template:Math $(\genfrac{}{}{0ex}{}{n}{k})$ such subsets (the binomial coefficient is zero for Template:Math). Therefore the right-hand side of (Template:EquationNote) evaluated at Template:Mvar equals

$\sum _{k=0}^m(\genfrac{}{}{0ex}{}{n}{k})\sum _{l=0}^k(-1{)}^{k-l}(\genfrac{}{}{0ex}{}{k}{l})c_l=\sum _{l=0}^m\underset{=:(*)}{\underbrace{\sum _{k=l}^n(-1{)}^{k-l}(\genfrac{}{}{0ex}{}{n}{k})(\genfrac{}{}{0ex}{}{k}{l})}}{c}_l,$

where we used that the first binomial coefficient is zero for Template:Math. Note that the sum (*) is empty and therefore defined as zero for Template:Math. Using the factorial formula for the binomial coefficients, it follows that

$\begin{array}{cc}(*)& =\sum _{k=l}^n(-1{)}^{k-l}\frac{n!}{k!(n-k)!}\frac{k!}{l!(k-l)!}\\ & =\underset{=(\genfrac{}{}{0ex}{}{n}{l})}{\underbrace{\frac{n!}{l!(n-l)!}}}\underset{=:(**)}{\underbrace{\sum _{k=l}^n(-1{)}^{k-l}\frac{(n-l)!}{(n-k)!(k-l)!}}}\\ \end{array}$

Rewriting (**) with the summation index Template:Math und using the binomial formula for the third equality shows that

$\begin{array}{cc}(**)& =\sum _{j=0}^{n-l}(-1{)}^j\frac{(n-l)!}{(n-l-j)!j!}\\ & =\sum _{j=0}^{n-l}(-1{)}^j(\genfrac{}{}{0ex}{}{n-l}{j})=(1-1{)}^{n-l}={\delta }_{ln},\end{array}$

which is the Kronecker delta. Substituting this result into the above formula and noting that Template:Mvar choose Template:Mvar equals Template:Math for Template:Math, it follows that the right-hand side of (Template:EquationNote) evaluated at Template:Mvar also reduces to Template:Math.

Representation in the polynomial ring

As a special case, take for Template:Mvar the polynomial ring Template:Math with the indeterminate Template:Mvar. Then (Template:EquationNote) can be rewritten in a more compact way as

This is an identity for two polynomials whose coefficients depend on Template:Mvar, which is implicit in the notation.

Proof of (Template:EquationNote) using (Template:EquationNote): Substituting Template:Math for Template:Math into (Template:EquationNote) and using the binomial formula shows that

$\sum _{n=0}^m{1}_{\{N=n\}}{x}^n=\sum _{k=0}^m{N}_k\underset{=(x-1{)}^k}{\underbrace{\sum _{l=0}^k(\genfrac{}{}{0ex}{}{k}{l})(-1{)}^{k-l}{x}^l}},$

which proves (Template:EquationNote).

Representation with shift and difference operators

Consider the linear shift operator Template:Mvar and the linear difference operator Template:Math, which we define here on the sequence space of Template:Mvar by

$\begin{array}{cc}E:V^{{\mathbb{N}}_0}& \to V^{{\mathbb{N}}_0},\\ E(c_0,c_1,c_2,c_3,\dots )& \mapsto (c_1,c_2,c_3,\dots ),\\ \end{array}$

and

$\begin{array}{cc}\mathrm{\Delta }:V^{{\mathbb{N}}_0}& \to V^{{\mathbb{N}}_0},\\ \mathrm{\Delta }(c_0,c_1,c_2,c_3\dots )& \mapsto (c_1-c_0,c_2-c_1,c_3-c_2,\dots ).\\ \end{array}$

Substituting Template:Math in (Template:EquationNote) shows that

where we used that Template:Math with Template:Mvar denoting the identity operator. Note that Template:Math and Template:Math equal the identity operator Template:Mvar on the sequence space, Template:Math and Template:Math denote the Template:Mvar-fold composition.

Let Template:Math denote the 0th component of the Template:Mvar-fold composition Template:Math applied to Template:Math, where Template:Math denotes the identity. Then (Template:EquationNote) can be rewritten in a more compact way as

Probabilistic versions

Consider arbitrary events Template:Math in a probability space Template:Math and let Template:Math denote the expectation operator. Then Template:Mvar from (Template:EquationNote) is the random number of these events which occur simultaneously. Using Template:Math from (Template:EquationNote), define

where the intersection over the empty index set is again defined as Template:Math, hence Template:Math. If the ring Template:Mvar is also an algebra over the real or complex numbers, then taking the expectation of the coefficients in (Template:EquationNote) and using the notation from (Template:EquationNote),

in Template:Math. If Template:Mvar is the field of real numbers, then this is the probability-generating function of the probability distribution of Template:Mvar.

Similarly, (Template:EquationNote) and (Template:EquationNote) yield

and, for every sequence Template:Math,

The quantity on the left-hand side of (Template:EquationNote) is the expected value of Template:Math.

Remarks

1. In actuarial science, the name Schuette–Nesbitt formula refers to equation (Template:EquationNote), where Template:Mvar denotes the set of real numbers.
2. The left-hand side of equation (Template:EquationNote) is a convex combination of the powers of the shift operator Template:Mvar, it can be seen as the expected value of random operator Template:Math. Accordingly, the left-hand side of equation (Template:EquationNote) is the expected value of random component Template:Math. Note that both have a discrete probability distribution with finite support, hence expectations are just the well-defined finite sums.
3. The probabilistic version of the inclusion–exclusion principle can be derived from equation (Template:EquationNote) by choosing the sequence Template:Math: the left-hand side reduces to the probability of the event Template:Math, which is the union of Template:Math, and the right-hand side is Template:Math, because Template:Math and Template:Math for Template:Math.
4. Equations (Template:EquationNote), (Template:EquationNote), (Template:EquationNote) and (Template:EquationNote) are also true when the shift operator and the difference operator are considered on a subspace like the [[Lp space#.E2.84.93p spaces|Template:Math spaces]].
5. If desired, the formulae (Template:EquationNote), (Template:EquationNote), (Template:EquationNote) and (Template:EquationNote) can be considered in finite dimensions, because only the first Template:Math components of the sequences matter. Hence, represent the linear shift operator Template:Mvar and the linear difference operator Template:Math as mappings of the Template:Math-dimensional Euclidean space into itself, given by the Template:Math-matrices

$E=\left(\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \ddots & ⋮\\ ⋮& \ddots & \ddots & \ddots & 0\\ 0& \cdots & 0& 0& 1\\ 0& \cdots & 0& 0& 0\end{array}\right),\mathrm{\Delta }=\left(\begin{array}{ccccc}-1& 1& 0& \cdots & 0\\ 0& -1& 1& \ddots & ⋮\\ ⋮& \ddots & \ddots & \ddots & 0\\ 0& \cdots & 0& -1& 1\\ 0& \cdots & 0& 0& -1\end{array}\right),$

and let Template:Mvar denote the Template:Math-dimensional identity matrix. Then (Template:EquationNote) and (Template:EquationNote) hold for every vector Template:Math in Template:Math-dimensional Euclidean space, where the exponent Template:Math in the definition of Template:Mvar denotes the transpose.

1. Equations (Template:EquationNote) and (Template:EquationNote) hold for an arbitrary linear operator Template:Mvar as long as Template:Math is the difference of Template:Mvar and the identity operator Template:Mvar.
2. The probabilistic versions (Template:EquationNote), (Template:EquationNote) and (Template:EquationNote) can be generalized to every finite measure space.

For textbook presentations of the probabilistic Schuette–Nesbitt formula (Template:EquationNote) and their applications to actuarial science, cf. Template:Harvtxt. Chapter 8, or Template:Harvtxt, Chapter 18 and the Appendix, pp. 577–578.

History

For independent events, the formula (Template:EquationNote) appeared in a discussion of Robert P. White and T.N.E. Greville's paper by Donald R. Schuette and Cecil J. Nesbitt, see Template:Harvtxt. In the two-page note Template:Harvtxt, Hans U. Gerber, called it Schuette–Nesbitt formula and generalized it to arbitrary events. Christian Buchta, see Template:Harvtxt, noticed the combinatorial nature of the formula and published the elementary combinatorial proof of (Template:EquationNote).

Cecil J. Nesbitt, PhD, F.S.A., M.A.A.A., received his mathematical education at the University of Toronto and the Institute for Advanced Study in Princeton. He taught actuarial mathematics at the University of Michigan from 1938 to 1980. He served the Society of Actuaries from 1985 to 1987 as Vice-President for Research and Studies. Professor Nesbitt died in 2001. (Short CV taken from Template:Harvtxt, page xv.)

Donald Richard Schuette was a PhD student of C. Nesbitt, he later became professor at the University of Wisconsin–Madison.

The probabilistic version of the Schuette–Nesbitt formula (Template:EquationNote) generalizes much older formulae of Waring, which express the probability of the events Template:Math and Template:Math in terms of Template:Math, Template:Math, ..., Template:Math. More precisely, with $(\genfrac{}{}{0ex}{}{k}{n})$ denoting the binomial coefficient,

and

see Template:Harvtxt, Sections IV.3 and IV.5, respectively.

To see that these formulae are special cases of the probabilistic version of the Schuette–Nesbitt formula, note that by the binomial theorem

${\mathrm{\Delta }}^k=(E-I{)}^k=\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}{E}^j,k\in {\mathbb{N}}_0.$

Applying this operator identity to the sequence Template:Math with Template:Mvar leading zeros and noting that Template:Math if Template:Math and Template:Math otherwise, the formula (Template:EquationNote) for Template:Math follows from (Template:EquationNote).

Applying the identity to Template:Math with Template:Mvar leading zeros and noting that Template:Math if Template:Math and Template:Math otherwise, equation (Template:EquationNote) implies that

$\mathbb{P}(N\ge n)=\sum _{k=n}^m{S}_k\sum _{j=n}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}.$

Expanding Template:Math using the binomial theorem and using equation (11) of the formulas involving binomial coefficients, we obtain

$\sum _{j=n}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}=-\sum _{j=0}^{n-1}(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}=(-1{)}^{k-n}(\genfrac{}{}{0ex}{}{k-1}{n-1}).$

Hence, we have the formula (Template:EquationNote) for Template:Math.

Applications

In actuarial science

Problem: Suppose there are Template:Mvar persons aged Template:Math with remaining random (but independent) lifetimes Template:Math. Suppose the group signs a life insurance contract which pays them after Template:Mvar years the amount Template:Math if exactly Template:Mvar persons out of Template:Mvar are still alive after Template:Mvar years. How high is the expected payout of this insurance contract in Template:Mvar years?

Solution: Let Template:Math denote the event that person Template:Mvar survives Template:Mvar years, which means that Template:Math. In actuarial notation the probability of this event is denoted by Template:Math and can be taken from a life table. Use independence to calculate the probability of intersections. Calculate Template:Math and use the probabilistic version of the Schuette–Nesbitt formula (Template:EquationNote) to calculate the expected value of Template:Math.

In probability theory

Let Template:Mvar be a random permutation of the set Template:Math and let Template:Math denote the event that Template:Mvar is a fixed point of Template:Mvar, meaning that Template:Math. When the numbers in Template:Mvar, which is a subset of Template:Math, are fixed points, then there are Template:Math ways to permute the remaining Template:Math numbers, hence

$\mathbb{P}\left(\bigcap _{j\in J}{A}_j\right)=\frac{(m-|J|)!}{m!}.$

By the combinatorical interpretation of the binomial coefficient, there are $(\genfrac{}{}{0ex}{}{m}{k})$ different choices of a subset Template:Mvar of Template:Math with Template:Mvar elements, hence (Template:EquationNote) simplifies to

$S_k=(\genfrac{}{}{0ex}{}{m}{k})\frac{(m-k)!}{m!}=\frac{1}{k!}.$

Therefore, using (Template:EquationNote), the probability-generating function of the number Template:Mvar of fixed points is given by

$𝔼[x^N]=\sum _{k=0}^m\frac{(x-1{)}^k}{k!},x\in ℝ.$

This is the partial sum of the infinite series giving the exponential function at Template:Math, which in turn is the probability-generating function of the Poisson distribution with parameter Template:Math. Therefore, as Template:Mvar tends to infinity, the distribution of Template:Mvar converges to the Poisson distribution with parameter Template:Math.

See also

• Rencontres numbers

References

• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation

External links

• Template:Mathgenealogy
• Template:Mathgenealogy