Let V denote a vector space over a field R such as the real or complex numbers (or more generally a module over a ring R with multiplicative identity).
Substituting this result into the above formula and noting that n choose l equals 1 for l = n, it follows that the right-hand side of (3) evaluated at ω also reduces to cn.
Then (3) can be rewritten in a more compact way as This is an identity for two polynomials whose coefficients depend on ω, which is implicit in the notation.
Proof of (4) using (3): Substituting cn = xn for n ∈ {0, ..., m} into (3) and using the binomial formula shows that which proves (4).
Note that E0 and Δ0 equal the identity operator I on the sequence space, Ek and Δk denote the k-fold composition.
To prove (5), we first want to verify the equation involving indicator functions of the sets A1, ..., Am and their complements with respect to Ω.
Then (3) can be rewritten in a more compact way as Consider arbitrary events A1, ..., Am in a probability space (Ω, F,
If the ring R is also an algebra over the real or complex numbers, then taking the expectation of the coefficients in (4) and using the notation from (7), in R[x].
If R is the field of real numbers, then this is the probability-generating function of the probability distribution of N. Similarly, (5) and (6) yield and, for every sequence c = (c0, c1, c2, c3, ..., cm, ...), The quantity on the left-hand side of (6') is the expected value of cN.
For textbook presentations of the probabilistic Schuette–Nesbitt formula (6') and their applications to actuarial science, cf.
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.
Donald Richard Schuette was a PhD student of C. Nesbitt, he later became professor at the University of Wisconsin–Madison.
Applying the identity to c = (0, ..., 0, 1, 1, 1, ...) with n leading zeros and noting that (E jc)0 = 1 if j ≥ n and (E jc)0 = 0 otherwise, equation (6') implies that Expanding (1 – 1)k using the binomial theorem and using equation (11) of the formulas involving binomial coefficients, we obtain Hence, we have the formula (9) for {N ≥ n}.
Problem: Suppose there are m persons aged x1, ..., xm with remaining random (but independent) lifetimes T1, ..., Tm.
In actuarial notation the probability of this event is denoted by t pxj and can be taken from a life table.
ways to permute the remaining m – |J| numbers, hence By the combinatorical interpretation of the binomial coefficient, there are