Schuette–Nesbitt formula - AbsoluteAstronomy.com

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, the Schuette–Nesbitt formula is a generalization of the probabilistic version of the inclusion-exclusion principle

Inclusion-exclusion principle

In combinatorics, the inclusion–exclusion principle is an equation relating the sizes of two sets and their union...

. It is named after Donald R. Schuette and Cecil J. Nesbitt

Cecil J. Nesbitt

Cecil James Nesbitt, Ph.D., F.S.A., M.A.A.A. was a mathematician who was a Ph.D. student of Richard Brauer and wrote many influential papers in the early history of modular representation theory. He taught actuarial mathematics at the University of Michigan from 1938 to 1980. Nesbitt was born in...

. The Schuette–Nesbitt formula

Formula

In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....

has practical applications in actuarial science

Actuarial science

Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in the insurance and finance industries. Actuaries are professionals who are qualified in this field through education and experience...

, where it is used to calculate the net single premium

Net premium valuation

A Net Premium Valuation is an actuarial calculation, used to place a value on the liabilities of a life insurer.-Background:It involves calculating a present value for the contractual liabilities of a contract, and deducting the value of future premiums. Both contractual liabilities, and future...

for life annuities

Life annuity

A life annuity is a financial contract in the form of an insurance product according to which a seller — typically a financial institution such as a life insurance company — makes a series of future payments to a buyer in exchange for the immediate payment of a lump sum or a series...

and life insurance

Life insurance

Life insurance is a contract between an insurance policy holder and an insurer, where the insurer promises to pay a designated beneficiary a sum of money upon the death of the insured person. Depending on the contract, other events such as terminal illness or critical illness may also trigger...

s based on the general symmetric status.

Statement of the formula

Consider arbitrary events A_{1, ..., A_m in a probability spaceProbability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...

$\scriptstyle(\Omega,\mathcal{F},\mathbb P)$ and let
$N=\sum_{n=1}^m 1_{A_n}$

denote the random numberRandom variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

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 intersectionIntersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

over the empty index setEmpty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

is defined as Ω, hence S₀ = 1.

Furthermore, consider the shift operatorShift operator
In mathematics, and in particular functional analysis, the shift operator or translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator....

E and the difference operator Δ, which we define here on the sequence spaceSequence space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers...

of a realReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

or complexComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

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 valueExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

of c_N.
Remarks
NEWLINENEWLINENormally, the name Schuette–Nesbitt formula refers to equation (**), where V denotes the set of real numbers.
NEWLINEE⁰ and Δ⁰ denote the identity operator I on the sequence space, Eⁿ and Δⁿ denote the n-fold compositionFunction composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...

.
NEWLINEIn equation (**), (Δⁿc)₀ denotes the 0th component of the sequence Δⁿc.
NEWLINEThe left-hand side of equation (*) is a convex combinationConvex combination
In convex geometry, a convex combination is a linear combination of points where all coefficients are non-negative and sum up to 1....

of the powers of the shift operator E, it can be seen as the expected valueExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

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.
NEWLINEThe 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 A_{1, ..., 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.}
NEWLINEEquations (*) and (**) are also true when the shift operator and the difference operator are considered on a subspace like the ℓ^p spaces.
NEWLINEIf 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 spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

into itself, given by the (m + 1) × (m + 1)-matricesMatrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

NEWLINE
NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINE $E=\begin{pmatrix}$
NEWLINE
NEWLINENEWLINE
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},

NEWLINENEWLINENEWLINENEWLINEand let I denote the (m + 1)-dimensional identity matrixIdentity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

. Then (**) holds for every vectorVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

c = (c₀, c₁, ..., c_m)^T in (m + 1)-dimensional Euclidean space, where the exponent T in the definition of c denotes the transposeTranspose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...

.
NEWLINENEWLINE

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 independentStatistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...

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. NesbittCecil J. Nesbitt
Cecil James Nesbitt, Ph.D., F.S.A., M.A.A.A. was a mathematician who was a Ph.D. student of Richard Brauer and wrote many influential papers in the early history of modular representation theory. He taught actuarial mathematics at the University of Michigan from 1938 to 1980. Nesbitt was born in...

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 proofCombinatorial proof
In mathematics, the term combinatorial proof is often used to mean either of two types of proof of an identity in enumerative combinatorics that either states that two sets of combinatorial configurations, depending on one or more parameters, have the same number of elements , or gives a formula...

; see the references below.

Cecil J. Nesbitt, PhDPHD
PHD may refer to:*Ph.D., a doctorate of philosophy*Ph.D. , a 1980s British group*PHD finger, a protein sequence*PHD Mountain Software, an outdoor clothing and equipment company*PhD Docbook renderer, an XML renderer...

, F.S.A., M.A.A.A., received his mathematical education at the University of TorontoUniversity of Toronto
The University of Toronto is a public research university in Toronto, Ontario, Canada, situated on the grounds that surround Queen's Park. It was founded by royal charter in 1827 as King's College, the first institution of higher learning in Upper Canada...

and the Institute for Advanced StudyInstitute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...

in PrincetonPrinceton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...

. He taught actuarial mathematics at the University of MichiganUniversity of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...

from 1938 to 1980. He served the Society of ActuariesSociety of Actuaries
The Society of Actuaries is a professional organization for actuaries based in North America. It was founded in 1949 as the merger of two major actuarial organizations in the United States: the Actuarial Society of America and the American Institute 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–MadisonUniversity of Wisconsin–Madison
The University of Wisconsin–Madison is a public research university located in Madison, Wisconsin, United States. Founded in 1848, UW–Madison is the flagship campus of the University of Wisconsin System. It became a land-grant institution in 1866...

.

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 coefficientBinomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

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

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 theoremBinomial theorem
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

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

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
$\sum_{j=k}^n\binom nj(-1)^{n-j}$ 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 A_{1, ..., A_m and their complementsComplement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...

with respect to Ω. Suppose an ω from Ω belongs to exactly k events out of A_{1, ..., A_m, where 0 ≤ k ≤ m, for simplicity of notation say that ω only belongs to A_{1, ..., 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
$1_{A_j^{\mathrm c}}I +1_{A_j}E1_\Omega I +1_{A_j}\Delta,\qquad j\in\{0,\ldots,m\}.$

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 expectationExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

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 notationActuarial notation
Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables.Traditional notation uses a halo system where symbols are placed as superscript or subscript before or after the main letter...

the probability of this event is denoted by ${}_tp_{x_j}$ and can be taken from a life tableLife table
In actuarial science, a life table is a table which shows, for each age, what the probability is that a person of that age will die before his or her next birthday...

. 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 theoremBinomial theorem
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

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

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 functionProbability-generating function
In probability theory, the probability-generating function of a discrete random variable is a power series representation of the probability mass function of the random variable...

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 permutationRandom permutation
A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. The use of random permutations is often fundamental to fields that use randomized algorithms such as coding theory, cryptography, and simulation...

of the set {1,...,m} and let A_j denote the event that j is a fixed pointFixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

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 coefficientBinomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

, 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 functionExponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

at z − 1, which in turn is the probability-generating function of the Poisson distributionPoisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...

with parameter 1. Therefore, as m tends to infinityInfinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

, the distribution of N converges to the Poisson distribution with parameter 1.

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.}}}}