Survival Functions and Hazard Rates for Exam P

Survival Function

The survival function S(x) = P(X > x) = 1 - F(x) gives the probability that a random variable exceeds x. For lifetime data, S(t) represents the probability of surviving beyond time t. Properties: S(0) = 1, S(infinity) = 0, and S is non-increasing. The PDF can be recovered as f(x) = -S'(x).

A useful identity for computing the mean of a non-negative random variable is E[X] = integral from 0 to infinity of S(x) dx. This avoids integration by parts and is often the fastest method on Exam P.

Hazard Rate (Force of Mortality)

The hazard rate (or force of mortality in life contingencies) is h(x) = f(x) / S(x) = -d/dx ln(S(x)). It represents the instantaneous rate of failure at time x given survival to time x. The cumulative hazard function is H(x) = integral from 0 to x of h(t) dt = -ln(S(x)), so S(x) = exp(-H(x)).

For the exponential distribution, h(x) = lambda (constant). For the Weibull with S(x) = exp(-(x/theta)^tau), h(x) = (tau/theta) * (x/theta)^(tau - 1), which increases when tau > 1 and decreases when tau < 1.

Exam P Applications

Exam P problems may specify a hazard rate and ask you to find the survival function, the PDF, or the expected lifetime. Given h(x), integrate to get H(x) = integral of h, then S(x) = exp(-H(x)). For example, if h(x) = 2x for x > 0, then H(x) = x^2 and S(x) = exp(-x^2), which is a Rayleigh-type distribution. The mean is integral from 0 to infinity of exp(-x^2) dx = sqrt(pi)/2.