Math Life Contingent Risk Models

Different formulas for force of mortality

Probability Tx <= dx

fx(t)=

Derivation of the above formula

Force of mortality at x+t formula

Integral from 0 to t of u x+s and Sx(t) formula

E[Tx²]

E[Tx]

Integral from 0 to t of So(t)

Capped future lifetime Random variable definition

E[min(Tx,n)]

l sub (x+t)

lx*(tPx)

tPx in terms of lx

l(x+t)/lx

Sx(t) formula With So fraction and the formula in xPx notation

Fx(t) in terms of Fo()

fx(t) formula fraction from Sx(t) fraction formula

derivation of book’s definition of force of mortality with limits and dx

Sx(t) e^ formula

formula when u sub x+s is constant over (0,t)

Fo(t) and So(t) and ux for De-Moivre’s distribution

Same as above for Tx

integral of a^x

a^x over ln(a)

So(t) and ux for Makehams’ distribution

So(t) and ux for Gompertz distribution

deferred mortality probability in q notation? Tx is between what and what? What is deferred mortality probability yin terms of survival function with x u , and t

(t+u)P(x) formula

tPx and tQx relation

They add to 1.

So(x) in xPx notation

xP0 (used to describe ux)

fx(t) formula using xPx function

Deriving Fx(t) from above

what happens to sPx when qx is very small

if qx is very small, probability of death within a year is close to 0 and probability of surviving at least a year is close to 1 and can be dropped from the integral, s is an interval from 0 to 1 and the midpoint is ½

fx(t) formula

E[Tx] or ex formula

Expectation of life at birth

E[Tx²]

Expectation of the capped future lifetime random variable

Curtate Lifetime definition

5 different forms for P(Kx=k)

Expected Curtate Life

Second moment of Kx

Capped Future Kurtate Lifetime expectation min(kx, N)

Rx residual part definition

on average how is curtate expectation of life related to normal expectation of life?

tPx and tQx in terms of lx

ndx and dx

nqx using the above formula

ndx/lx

qx using above formula

dx/lx

force of mortality from lx

lx and l0 formula derived from survival function relationship

xP0 is equal to?

Integral from 0 to infinity of xP0 is what?

E[T0]

previous flashcard equal to what other integral?

Curtate expectation in terms of lx

l(x+t) in terms of dx

what is tqx when t is between 0 and 1 and it is uniform in the year

From u(x+t) formula involving lx+t, derive the u(x+t) formula involving qx

Under UDD fx(t)=

qx for t between 0 and 1

What typical formulas do not exist under UDD?

Under exponential interpolation what is l(x+t) in l(x+1) and lx

Under exponential interpolation what is lx+t in terms of tPx

derivative of (Px)^t

ln(Px)*(Px)^t

force of mortality at x+t under exponential model

Denotes u*

what is tPx under this model using exponents?

and fx(t) is equal to what? this isn exponential distribution with what parameter?

u* as lambda

and under memoryless property what is tPx+s

what is 1 over l(x+t) in harmonic interpolation?

l(x+t) in the harmonic model

what is (1-t)q(x+t) in harmonic model

(1-t)*qx

relationships between T(x) and T(x+n) alongside tP([x+n])