Schuette–Nesbitt formula

In probability theory, the Schuette–Nesbitt formula is a generalization of the probabilistic version of the inclusion-exclusion principle. It is named after Donald R. Schuette and Cecil J. Nesbitt.

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.

Statement of the formula

Consider arbitrary events A₁, ..., A_m in a probability space $\scriptstyle(\Omega,\mathcal{F},\mathbb P)$ and let

$N=\sum_{n=1}^m 1_{A_n}$

denote the random number of these events which occur simultaneously. Define

$S_n=\sum_{\scriptstyle J\subset\{1,\ldots,m\}\atop\scriptstyle|J|=n} \mathbb{P}\biggl(\bigcap_{j\in J}A_j\biggr),\qquad n\in\{0,\ldots,m\},$

where the intersection over the empty index set is defined as Ω, hence S₀ = 1.

Furthermore, consider the shift operator E and the difference operator Δ, which we define here on the sequence space of a real or complex vector space V by

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

and

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

Then

$\sum_{n=0}^m \mathbb{P}(N=n)E^n=\sum_{n=0}^m S_n\Delta^n\qquad(*)$

and, for every sequence c = (c₀, c₁, c₂, c₃, ..., c_m, ...),

$\sum_{n=0}^m c_n\,\mathbb{P}(N=n)=\sum_{n=0}^m S_n\,(\Delta^nc)_0.\qquad(**)$

The quantity in (**) is the expected value of c_N.

Remarks

1. Normally, the name Schuette–Nesbitt formula refers to equation (**), where V denotes the set of real numbers.
2. E⁰ and Δ⁰ denote the identity operator I on the sequence space, Eⁿ and Δⁿ denote the n-fold composition.
3. In equation (**), (Δⁿc)₀ denotes the 0th component of the sequence Δⁿc.
4. The left-hand side of equation (*) is a convex combination of the powers of the shift operator E, it can be seen as the expected value of random operator E^N. Accordingly, the left-hand side of equation (**) is the expected value of random component c_N. Note that both have a discrete distribution, hence expectations are just the well-defined finite sums.
5. The inclusion-exclusion principle can be derived from equation (**) by choosing the sequence c = (0,1,1,...): the left-hand side reduces to the probability of the event {N ≥ 1}, which is the union of A₁, ..., A_m, and the right-hand side is S₁ – S₂ + S₃ – ... – (–1)^mS_m, because (Δ⁰c)₀ = 0 and (Δⁿc)₀ =  –(–1)ⁿ for 1 ≤ n ≤ m.
6. Equations (*) and (**) are also true when the shift operator and the difference operator are considered on a subspace like the ℓ ^p spaces.
7. If desired, the formulae and the proof below can be considered in finite dimensions, because only the first m + 1 components of the sequences matter. Hence, represent the shift operator E and the difference opertor Δ as mappings of the (m + 1)-dimensional Euclidean space into itself, given by the (m + 1) × (m + 1)-matrices

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

and let I denote the (m + 1)-dimensional identity matrix. Then (**) holds for every vector c = (c₀, c₁, ..., c_m)^T in (m + 1)-dimensional Euclidean space, where the exponent T in the definition of c denotes the transpose.

For textbook presentations of the Schuette–Nesbitt formula and their applications to actuarial science, cf. Gerber Life Insurance Mathematics, Chapter 8, or Bowers et al. Actuarial Mathematics, Chapter 18 and the Appendix, pp. 577–578.

History

For independent events, the formula appeared in 1959 in a discussion of Robert P. White and T.N.E. Greville's paper by Donald R. Schuette and Cecil J. Nesbitt in the Transactions of Society of Actuaries. In a two-page note appearing 1979, Hans U. Gerber called it Schuette–Nesbitt formula and generalized it to arbitrary events. In 1994, Christian Buchta published an elementary combinatorial proof; see the references below.

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 Bowers et al., page xv.)

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

The Schuette–Nesbitt formula (**) generalizes much older formulae of Waring, which express the probability of the events {N = k} and {N ≥ k} in terms of S₁, S₂, ..., S_m. More precisely, with $\textstyle\binom nk$ denoting the binomial coefficient,

$\mathbb{P}(N=k)=\sum_{n=k}^m(-1)^{n-k}\binom nk S_n, \qquad k\in\{0,\ldots,m\},$

and

$\mathbb{P}(N\ge k)=\sum_{n=k}^m(-1)^{n-k}\binom{n-1}{k-1}S_n,\qquad k\in\{1,\ldots,m\},$

see Feller, Sections IV.3 and IV.5, respectively.

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

$\Delta^n=(E-I)^n=\sum_{j=0}^n\binom nj (-1)^{n-j}E^j,\qquad n\in\mathbb{N}_0.$

Applying this operator identity to the sequence c = (0, ..., 0, 1, 0, 0, ...) with k leading zeros and noting that (E ^jc)₀ = 1 if j = k and (E ^jc)₀ = 0 otherwise, the first formula for {N = k} follows from (**).

Applying the identity to c = (0, ..., 0, 1, 1, 1, ...) with k leading zeros and noting that (E ^jc)₀ = 1 if j ≥ k and (E ^jc)₀ = 0 otherwise, equation (**) implies that

$\mathbb{P}(N\ge k)=\sum_{n=k}^m S_n\sum_{j=k}^n\binom nj(-1)^{n-j}.$

Expanding (1 − 1)ⁿ using the binomial theorem and using equation (11) of the formulas involving binomial coefficients, we obtain

$\sum_{j=k}^n\binom nj(-1)^{n-j} =-\sum_{j=0}^{k-1}\binom nj(-1)^{n-j} =(-1)^{n-k}\binom{n-1}{k-1}.$

Hence, we have the formula for {N ≥ k}

Proof

The second equation (**) follows from the first one (*) by applying it to the sequence c and considering of the 0th component, because (Eⁿc)₀ = c_n.

To prove (*), we first want to verify the operator equation

$\sum_{n=0}^m 1_{\{N=n\}}E^n=\prod_{j=1}^m(1_{A_j^{\mathrm c}}I +1_{A_j}E)$

involving indicator functions of the events A₁, ..., A_m and their complements with respect to Ω. Suppose an ω from Ω belongs to exactly k events out of A₁, ..., A_m, where 0 ≤ k ≤ m, for simplicity of notation say that ω only belongs to A₁, ..., A_k. Then the left-hand side is E^k. On the right-hand side, the first k factors equal E, the remaining ones equal the identity operator I, their product is also E^k, hence the formula is true.

Note that the difference operator Δ is the difference of the shift operator E and the identity operator I, meaning that Δ = E – I, hence

$1_{A_j^{\mathrm c}}I +1_{A_j}E =1_\Omega I-1_{A_j}I +1_{A_j}E =1_\Omega I +1_{A_j}\Delta,\qquad j\in\{0,\ldots,m\}.$

Inserting this result into the operator equation and expanding the product gives

$\sum_{n=0}^m 1_{\{N=n\}}E^n =\sum_{n=0}^m\sum_{\scriptstyle J\subset\{1,\ldots,m\}\atop\scriptstyle|J|=n} 1_{\cap_{j\in J}A_j}\Delta^n,$

because the product of indicator functions is the indicator function of the intersection.

Taking the expectation and using its linearity, we get (*).

An application in actuarial science

Problem: Suppose there are m persons aged x₁, ..., x_m with remaining random (but independent) lifetimes T₁, ..., T_m. Suppose the group signs a life insurance contract which pays them after t years the amount c_n if exactly n persons out of m are still alive after t years. How high is the expected payout of this insurance contract in t years?

Solution: Let A_j denote the event that person j survives t years, which means that A_j = {T_j > t}. In actuarial notation the probability of this event is denoted by ${}_tp_{x_j}$ and can be taken from a life table. Use independence to calculate the probability of intersections. Calculate S₁, ..., S_m and use the Schuette–Nesbitt formula (**) to calculate the expected value of c_N.

An application in probability theory

For a real number z set c_n = zⁿ for 0 ≤ n ≤ m. By the binomial theorem,

$\Delta^n=(E-I)^n=\sum_{k=0}^n\binom nk (-1)^{n-k}E^k,$

hence

$(\Delta^nc)_0=\sum_{k=0}^n\binom nk (-1)^{n-k}z^k=(z-1)^n.$

Using the Schuette–Nesbitt formula (**), we get for the probability-generating function of N

$\mathbb{E}[z^N]=\sum_{n=0}^m(z-1)^nS_n,\qquad z\in\mathbb{R}.$

Now let σ be a random permutation of the set {1,...,m} and let A_j denote the event that j is a fixed point of σ, meaning that A_j = {σ(j) = j}. When the numbers in J, which is a subset of {1,...,m}, are fixed points, then there are (m − |J|)! ways to permute the remaining m − |J| numbers, hence

$\mathbb{P}\biggl(\bigcap_{j\in J}A_j\biggr)=\frac{(m-|J|)!}{m!}.$

By the combinatorical interpretation of the binomial coefficient, there are $\textstyle\binom mn$ different choices of a subset J of {1,...,m} with n elements, hence

$S_n=\binom mn \frac{(m-n)!}{m!}=\frac1{n!}$

and

$\mathbb{E}[z^N]=\sum_{n=0}^m\frac{(z-1)^n}{n!},\qquad z\in\mathbb{R}.$

This is the partial sum of the infinite series giving the exponential function at z − 1, which in turn is the probability-generating function of the Poisson distribution with parameter 1. Therefore, as m tends to infinity, the distribution of N converges to the Poisson distribution with parameter 1.

See also

• Rencontres numbers

References

1. Newton L. Bowers, Hans U. Gerber, James C. Hickman, Donald A. Jones, Cecil J. Nesbitt: Actuarial Mathematics. (2nd ed.), 1997, The Society of Actuaries ISBN 0-938959-46-8.
2. Christian Buchta: An Elementary Proof of the Schuette–Nesbitt Formula. Mitteilungen der Schweiz. Vereinigung der Versicherungsmathematiker, Heft 2/1994, pages 219–220.
3. William Feller: An Introduction to Probability Theory and Its Applications, Volume I. (3rd ed., revised printing), 1968, John Wiley & Sons. ISBN 0-471-25708-7.
4. Hans U. Gerber: A Proof of the Schuette–Nesbitt Formula for Dependent Events. Actuarial Research Clearing House, 1979, Vol. 1. pages 9–10. Online
5. Hans U. Gerber: Life Insurance Mathematics. (3rd ed.), 1997, Springer-Verlag, ISBN 3-54062-242-X.
6. Donald R. Schuette and Cecil J. Nesbitt, Discussion of the preceding paper by Robert P. White and T.N.E. Greville, Transactions of Society of Actuaries, 1959, Vol. 11, No. 29AB, pages 97–99. Online

External links

• Cecil J. Nesbitt at the Mathematics Genealogy Project.
• Donald R. Schuette at the Mathematics Genealogy Project.