I have a empirical cumulative probability distribution function for a random variable. The random variable is "time to failure" and I have the full curve i.e till the probability reaches $1$. I want to know Mean Time To Failure i.e expectation of that random variable. Is there any standard method to find mean from an empirical distribution.
I am getting the empirical CDF (as discrete values) as output from a "model checking tool" which uses iterative numerical computation techniques to get those probabilities. For example, let $F(t)=P(X \leq t)$ is the CDF of the random variable $X$ where $X$ stands for time between failure. To plot the curve of "$F(t)$ vs $t$" I am varying t with some step size, calculating $F(t)$ for that t using the "model checking tool" and adding the points to get the curve. I can use small step size to get the more accurate curve. So, I have access to only this CDF values at different$ t$. From this values I want to do a good estimate of mean value of $X$.
Now the parameters will be:
1) $T$, the maximum value of t. We need to find this
with some precision i.e if $F(T_1)-F(T_2)$ is less than some epsilon we set $T=T_1$.
2) Once we have found T we need to find suitable step size $h$ at which we will be
calculating the CDF values.
How should I choose those parametrs?
Best Answer
Depending on how you've got your empirical cdf (discretized, formula) you might opt for Henning's answer, or also (given that the variable is positive) use this identity (obtained by applying integration by parts) :
$$E(X) = \int_0^\infty (1- F(x)) \; dx$$