# Mean Residual Time

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Join For Freeif 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.

Published at DZone with permission of John Cook, DZone MVB. See the original article here.

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