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 and subsets . Let

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

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

where denotes the indicator function of the set of all with , and $\textstyle\binom kl$ is a binomial coefficient. Equality () says that the two -valued functions defined on are the same.

Proof of ()

We prove that () holds pointwise. Take and define .
Then the left-hand side of () equals .
Let denote the set of all those indices such that , hence contains exactly indices.
Given with elements, then belongs to the intersection if and only if is a subset of .
By the combinatorial interpretation of the binomial coefficient, there are $\textstyle\binom nk$ such subsets (the binomial coefficient is zero for ).
Therefore the right-hand side of () evaluated at equals

:$\sum_^m \binom nk\sum_^k (-1)^\binom klc_l =\sum_^m\underbrace_ c_l,$

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

:$\begin (*) &=\sum_^n (-1)^\frac\,\frac\\ &=\underbrace_\underbrace_\\ \end$
Rewriting (**) with the summation index und using the binomial formula for the third equality shows that
:$\begin (**) &=\sum_^ (-1)^\frac\\ &=\sum_^ (-1)^\binom =(1-1)^ =\delta_, \end$
which is the Kronecker delta. Substituting this result into the above formula and noting that choose equals for , it follows that the right-hand side of () evaluated at also reduces to .

Representation in the polynomial ring

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

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

Proof of () using (): Substituting for into () and using the binomial formula shows that
:$\sum_^m 1_x^n =\sum_^m N_k\underbrace_,$
which proves ().

Representation with shift and difference operators

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

:$\begin E:V^&\to V^,\\ E(c_0,c_1,c_2,c_3,\ldots)&\mapsto(c_1,c_2,c_3,\ldots),\\ \end$

and

:$\begin \Delta:V^&\to V^,\\ \Delta(c_0,c_1,c_2,c_3\ldots)&\mapsto(c_1-c_0,c_2-c_1,c_3-c_2,\ldots).\\ \end$

Substituting in () shows that

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

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

Probabilistic versions

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

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

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

Similarly, () and () yield

and, for every sequence ,

The quantity on the left-hand side of () is the expected value of .

Remarks

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

:::$E=\begin 0&1&0&\cdots&0\\ 0&0&1&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&0\\ 0&\cdots&0&0&1\\ 0&\cdots&0&0&0 \end, \qquad \Delta=\begin -1&1&0&\cdots&0\\ 0&-1&1&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&0\\ 0&\cdots&0&-1&1\\ 0&\cdots&0&0&-1 \end,$

::and let denote the -dimensional identity matrix. Then () and () hold for every vector in -dimensional Euclidean space, where the exponent in the definition of denotes the transpose.
#Equations () and () hold for an arbitrary linear operator as long as is the difference of and the identity operator .
#The probabilistic versions (), () and () can be generalized to every finite measure space.

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

History

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

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 , 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 () generalizes much older formulae of Waring, which express the probability of the events and in terms of , , ..., . More precisely, with $\textstyle\binom kn$ denoting the binomial coefficient,

and

see , 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

:$\Delta^k=(E-I)^k=\sum_^k\binom kj (-1)^E^j,\qquad k\in\mathbb_0.$

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

Applying the identity to with leading zeros and noting that if and otherwise, equation () implies that

:$\mathbb(N\ge n)=\sum_^m S_k\sum_^k\binom kj(-1)^.$

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

:$\sum_^k\binom kj(-1)^ =-\sum_^\binom kj(-1)^ =(-1)^\binom.$

Hence, we have the formula () for .

An application in actuarial science

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

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

An application in probability theory

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

:$\mathbb\biggl(\bigcap_A_j\biggr)=\frac.$

By the combinatorical interpretation of the binomial coefficient, there are $\textstyle\binom mk$ different choices of a subset of with elements, hence () simplifies to

:$S_k=\binom mk \frac=\frac1.$

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

:\mathbb ^N\sum_^m\frac,\qquad x\in\mathbb.

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

See also

*Rencontres numbers

References

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External links

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{{DEFAULTSORT:Schuette-Nesbitt formula
Enumerative combinatorics
Probability theorems
Theorems in statistics
Articles containing proofs