Mean Residual Time

if something has survived this far, how much longer is it expected to survive? that’s the question answered by mean residual time.

for a positive random variable x , the mean residual time for x is a function e _x ( t ) given by

provided the expectation and integral converge. here f (t) is the cdf, the probability that x is greater than t .

for an exponential distribution, the mean residual time is constant. for a pareto (power law) distribution, the mean residual time is proportional to t . this has an interesting consequence, known as the lindy effect .

now let’s turn things around. given function a function e ( t ), can we find a density function for a positive random variable with that mean residual time? yes.

the equation above yields a differential equation for f , the cdf of the distribution.

if we differentiate both sides of

with respect to t and rearrange, we get the first order differential equation

where

the initial condition must be f (0) = 0 because we’re looking for the distribution of a positive random variable, i.e. the probability of x being less than zero must be 0. the solution is then

this means that for a desired mean residual time, you can use the equation above to create a cdf function to match. the derivative of the cdf function gives the pdf function, so differentiate both sides to get the density.