Fx(t) = P(Tx < t) = P(Tx < t)

(random variable Tx represents the future lifetime at age x; the sharp inequality when calculating the distribution function Fx(t) is justified by the continuity of the random variable Tx)

x + Tx (whole lifetime (past and future) of a life aged x) Sx(t) = P(Tx > t) (survival function at age x)

qx = Fx(1) = P(Tx < 1)

(probability of death at age x (see Sect. 17.2): is the probability that a life aged x will die within 1 year)

px = Sx(1) = P(Tx > 1)

(probability of survival modeling at age x (see Sect. 17.2): is the probability that a life aged x will survive to age x +1)

s\qx = Fx(s + 1) - Fx(s) = P(s < Tx < s + 1) (probability that a life aged x will die at age x + s; it simplifies to qx for s = 0)

tqx = Fx(t) = P(Tx < t)

(t-year probability of death at age x: is the probability that a life aged x will die within t years; it simplifies to qx for t = 1)

tpx = Sx(t) = P(Tx > t)

(t-year probability of survival modeling at age x: is the probability that a life aged x will survive to age x + t; it simplifies to px for t = 1)

s\tqx = Fx(s + t) - Fx(s) = P(s < Tx < s + t)

(probability that a life aged x will survive to age x + s, but will die within further t years; it simplifies to s\qx for t = 1)

s+tpx = ipx ¦ tpx+s

s+tqx = 1 - (1 - sqx) ¦ (1 - tqx+s) s|qx = ipx ¦ qx+s; s| tqx = ipx ¦ tqx+s

npx = px ¦ px+1 ¦ ••• ¦ px+n-1

Assumption: the random variable Tx (see thereinbefore) has probability density (see Sect. 26.3):

fx(t)=dF(t)=- bp*

ixx+t=fx^t) = fx(t) = -d ln(tpx) (force of mortality at age x + t of a life aged x)

tpx Sx(t) dt

ix ¦ Ax & P(Tx < Ax) = Axqx

(|x ¦ Ax can be interpreted as an approximation of the probability that a life aged x will die within age interval (x, x + Ax) of a (small) length Ax)

(values in life tables (see Sect. 17.2) calculated with respect to continuous  arguments)

Laws of mortality in Insurance

Represent mortality decrement orders expressed by means of analytical functions; they can be classified according to the corresponding force  of mortality:

• constant force ofmortality:

Xx = X; tpx = e-'.

• De Moivre's law of mortality: has Tx with the uniform distribution (see Sect.
  26.5):

1              m - x - t

Xx =              ; tpx =             

m - x              m - x

• Gompertz's law of mortality: has xx that increases exponentially:

Xx = B ¦ cx; tpx = gcx(ct-1) (B > 0; c > 1; g = exp{-B/ ln(c)})

• Gompertz-Makeham's law of mortality: modifies Gompertz's law:

txx = A + B • C; tPx = st • g^-1 (A > 0; B > 0; c > 1; g = exp{-B/ ln(c)}; s = exp(-A))

• Weibull's law of mortality: has /xx that increases polynomially:

^x = k•x"; tPx = w(x+t)n+1- x"+1 (k > 0; n > 0; w = exp{-k/(n+1)})

Tx = Kx + Sx, where Kx = [Tx] and0 < Sx < 1

(Kx is the curtate future lifetime at age x (i.e. a random variable with values 0, 1, 2,... representing the integer number of completed future years lived by a person aged x); Sx is the fraction of the year, during which the person aged x is alive in the year of death)

P(Kx = k) = kPx - k+1Px = kPx • qx+ k, k = 0,1,2,...

(X

ex = E(Kx) = Y^ k • P(Kx = k) = Y k kPx • qx+ k = ^ kPx k=0 k=1 k=1 (life exPectancy at age x as the expected curtate future lifetime at age x (in contrast to °ex = E(Tx)))

°ex « ex + 2

(approximation of E(Tx), given that Sx (i.e. the fraction of the year of death, see thereinbefore) has the uniform distribution (see Sect. 26.5) in interval (0, 1), see also Sect. 17.2)

X              / X              \ 2

i ^ 2              V i

var(Tx) « var(Kx) + — = k ¦ kPx • qx+ k - I kPx\ + —

k=i \k=i '

(approximation of var(Tx), given that Sx (i.e. the fraction of the year of death, see thereinbefore) has the uniform distribution (see Sect. 26.5) in interval (0, 1))

Some approximations of the distribution of Sx (i.e. the fraction of the year of death, see thereinbefore):

- assumPtion of linearity: tqx = t • qx for 0 < t < 1:

then (1) t px = 1 - t • qx for 0 < t < 1; (2) Kx and Sx are independent; (3) Sx has the uniform distribution in (0, 1) assumption of constant force of mortality: X x + t — X for 0 < t < 1:  then (1) t px = (px ). for 0 < t < 1; (2) Kx and Sx are not independent; (3) Sx  has the censored exponential distribution in (0, 1)  rounding of Sx to next forthcoming multiple of 1/m: Sx(m) = [m ¦ Sx + 1]/m:  then (1) if Kx and Sx are independent, then also Kx and Sx(m) are independent;  (2) if Sx has the uniform distribution in (0, 1), then also Sx(m) has the uniform  (but discrete) distribution in (0, 1)