Estimate remaining time to failure based only on average time to failure Suppose a random variable X representing time to failure. I don't know its pdf and only know its E[X]. How can I derive E[ X | X > t], i.e. average remaining time to failure? If it is impossible to do so without knowing its pdf, is it possible to come up with a decent estimate based on the knowledge of E[X] only. Human life expectancy is a concrete example of this problem. Given that the life expectancy = 75, is it possible to get a decent estimate for how much longer a person aged 80 can live?
A graph below is plotted on the data points sampled from X. It shows the shape of X density function. It always is left skewed and have a right long tail.

 A: I feel like I might need more detail, but I going to make some assumptions here. If they are wrong please correct me and I will change my answer or delete it so I do not confuse anybody.
When you say "shape of X", I assume you mean it is a density function. Therefore, you should first create an empirical survival function for the purpose of my explanation. Please note that there are many ways to approach this. I am trying to offer a very straight forward way of doing it. 
To create the survival curve, calculate 1 minus the empirical cumulative distribution function. Then you should have a graph (assuming no censoring) that starts at $S(0)=1$ and goes to $S(\infty)=0$ and is monotonically decreasing as "t" gets large.
So generally speaking the integral of the survival function is the actual mean survival time of the distribution. Therefore I would write out the conditional mean as follows:
$$E(T|T>t) = \int_{t}^{\infty} \frac{S(x)}{S(t)} dx $$
where $S(t) = P(T > t)$ is the unconditional survival curve. So just take the graph above and apply a numerical integration function to it from any "t" to $\infty$ (or pick a really large number as the integral's upper bound instead of using infinity) and divide this integral by the survival curve's value at "t" (i.e. S(t)).
Then store this value in a vector for each value of "t" and loop over as many values of "t" that you want and you have a nice graph of the conditional mean for every value of "t". If you wanted just an expected value for a single "t", then you are all set. Good luck!
