Chapter 7: Multiple Life Models
,1. The Joint Life Status
▶ In the previous chapters, Tx is the random variable representing (x)’s future
lifetime. It is also possible to regard the time until failure of a more general
status, where what is meant by a status can be an individual’s lifetime, or
the time until some event related to a group of individuals, or the
breakdown of a machine. Some examples are the time until the first death
of a pair of lives, or the time until the second death, or the time until
occurrence of some specified event. If we have a status w for which the
notion of failure is clearly defined and we denote the time until failure of w
by Tw . The analysis of Tw is similar to Tx with the following relationships:
t qw = P[Tw ≤ t] = Fw (t) is the probability of failure occurring by time t.
t pw = 1 − t qw = P[Tw > t] = Sw (t).
▶ In general, it is not true for all statuses that t pw = n pw · t−n pw +n (it is true
for some)
▶ We do have
t|u qw = t pw − t+u pw = t+u qw − t qw = P[t < Tw ≤ t + u],
but it is not always true that t|u qw = t pw · u qw +t .
,1. The Joint Life Status
d
▶ The “force of failure” at time t from status w is µw (t) = − dt pt pw . For a
t w
single life status w = (x), we have µx (t) = µx+t .
▶ The probability density function is fw (t) = t pw µw (t).
Rn Rn R∞
n qw = 0 t pwR µw (t)dt, n pw = exp[− 0 µw (t)dt], e̊w = 0 t pw dt.
∞
Var[Tw ] = 2 0 t · t pw dt − (e̊w )2 .
▶ Joint distributions of future lifetimes: given random variables Tx and Ty ,
there will be a density function of the joint distribution between Tx and
Ty and it is denoted by fx,y (u, v ). The joint distribution function is
Rt Rs
Fx,y (s, t) = P(Tx ≤ s, Ty ≤ t) = 0 0 fx,y (u, v )dudv and the joint
survival function is R∞R∞
Sx,y (s, t) = P(Tx > s, Ty > t) = t s fx,y (u, v )dudv .
▶ In general, Sx,y (s, t) ̸= 1 − Fx,y (s, t). This is contrast with the case for a
single life.
, 1. The Joint Life Status
▶ Each of Tx and Ty have marginal distributions and we have
s qx = Fx (s) = limt→∞ Fx,y (s, t) and
t qy = Fy (t) = lims→∞ Fx,y (s, t). We similarly have marginal
distributions of Kx and Ky .
▶ The covariance is defined as Cov[Tx , Ty ] = E[Tx · Ty ] − E[Tx ]E[Ty ]
Cov[Tx ,Ty ]
and the correlation coefficient is defined as ρx,y = √ .
Var[Tx ]Var[Ty ]
,1. The Joint Life Status
▶ In the previous chapters, Tx is the random variable representing (x)’s future
lifetime. It is also possible to regard the time until failure of a more general
status, where what is meant by a status can be an individual’s lifetime, or
the time until some event related to a group of individuals, or the
breakdown of a machine. Some examples are the time until the first death
of a pair of lives, or the time until the second death, or the time until
occurrence of some specified event. If we have a status w for which the
notion of failure is clearly defined and we denote the time until failure of w
by Tw . The analysis of Tw is similar to Tx with the following relationships:
t qw = P[Tw ≤ t] = Fw (t) is the probability of failure occurring by time t.
t pw = 1 − t qw = P[Tw > t] = Sw (t).
▶ In general, it is not true for all statuses that t pw = n pw · t−n pw +n (it is true
for some)
▶ We do have
t|u qw = t pw − t+u pw = t+u qw − t qw = P[t < Tw ≤ t + u],
but it is not always true that t|u qw = t pw · u qw +t .
,1. The Joint Life Status
d
▶ The “force of failure” at time t from status w is µw (t) = − dt pt pw . For a
t w
single life status w = (x), we have µx (t) = µx+t .
▶ The probability density function is fw (t) = t pw µw (t).
Rn Rn R∞
n qw = 0 t pwR µw (t)dt, n pw = exp[− 0 µw (t)dt], e̊w = 0 t pw dt.
∞
Var[Tw ] = 2 0 t · t pw dt − (e̊w )2 .
▶ Joint distributions of future lifetimes: given random variables Tx and Ty ,
there will be a density function of the joint distribution between Tx and
Ty and it is denoted by fx,y (u, v ). The joint distribution function is
Rt Rs
Fx,y (s, t) = P(Tx ≤ s, Ty ≤ t) = 0 0 fx,y (u, v )dudv and the joint
survival function is R∞R∞
Sx,y (s, t) = P(Tx > s, Ty > t) = t s fx,y (u, v )dudv .
▶ In general, Sx,y (s, t) ̸= 1 − Fx,y (s, t). This is contrast with the case for a
single life.
, 1. The Joint Life Status
▶ Each of Tx and Ty have marginal distributions and we have
s qx = Fx (s) = limt→∞ Fx,y (s, t) and
t qy = Fy (t) = lims→∞ Fx,y (s, t). We similarly have marginal
distributions of Kx and Ky .
▶ The covariance is defined as Cov[Tx , Ty ] = E[Tx · Ty ] − E[Tx ]E[Ty ]
Cov[Tx ,Ty ]
and the correlation coefficient is defined as ρx,y = √ .
Var[Tx ]Var[Ty ]
