# 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.

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