Chapter 3: Survival Distributions and Life Tables
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3.1 Survival Function
• Let X be the age-at-death of a life (ie. Newborn’s age at
death)
è X is a positive random variable (X ≥ 0)
• The cumulative distribution function (cdf) of X is
FX(x) = P(X = x) , x = 0
= Probability that the newborn will die before age x
The function FX(x) has the following properties:
1. FX(0) = 0
2. limx?8
FX(x) = 1
è If there is a limit age, we denote it by w (such as w =
100 for human survival).
è That means, FX(x) = 1, for all x = w.
3. FX(x) is a nondecreasing function
Note1: Properties (2) and (3) apply to any cdf.
Note2: Property (1) apply to any positive r.v..
Chapter 3: Survival Distributions and Life Tables
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Survival Function:
• Let s(x) be a survival function of X:
s(x) = 1 – FX(x) = 1 – P(X = x) = P(X > x)
We can deduce the properties of s(x) from those of FX(x).
1. s(0) = 1
2. limx?8
s(x) = 0
è If there is a limit age, then s(x) = 0, for all x = w.
3. s(x) is a nonincreasing function
P(x < X = z) = Probability that a newborn dies between
ages x and z (with x < z)
= FX(z) - FX(x)
= s(x) – s(z)
Example: Consider the cdf
0 = x = 100 .
a) Verify that this function satisfies the 3 properties stated
before.
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