Some Mathematics
of Reliability Theory – Survival Analysis
In this blog I am trying to
explain mathematical concepts with real life metaphors and examples. The
discussion on similarities between reliability theory and survival analysis is
concluded in today’s blog with explanation on mathematics behind this concept.
This mathematics summarizes the discussion conducted in previous five parts.
Denoting the life of a device/
human by a random variable T, t denotes the specific value taken by this random
variable.
R(t) = Reliability of device at
time t = P( T greater than equal to t) =
Probability that the life time of device is greater than t
S(t)= Survival of an individual
at least till time t = P( T greater than equal to t) = Probability that an
individual survives at time t
Cumulative distribution function =
F(t) = 1 - R(t) = Probability that the device fails before t
Cumulative distribution function =
F(t) = 1 - S(t) = Probability that an individual dies before t
h(t) = hazard rate = Probability (the device
fails between time t and t + Δ t|The device has survived till t)
= -R’(t)/R(t)
h(t)= instantaneous force of
mortality = Probability ( an individual dies between time t and t + Δ
t|The individual survived till t) = -S’(t)/S(t)
h(t) = λ=constant for electronic device – memory less property and
independent of time-Reliability Theory
h(t) = λ(t) = function of time for
human life – Survival Analysis