I'm trying to work out how to solve what seems like a simple problem, but I can't convince myself of the correct method.

I have time-series data that represents the pdf of a Power output (P), varying over time, also the cdf and quantile functions - f(P,t), F(P,t) and q(p,t). I need to find the pdf, cdf and quantile function for the Energy in a given time interval [t1,t2] from this data - say e(), E(), and qe().

Clearly energy is the integral of the power over [t1,t2], but how do I best calculate e, E and qe ?

My best guess is that since q(p,t) is a power, I should generate qe by integrating q over the time interval, and then calculate the other distributions from that.

Is it as simple as that, or do I need to get to grips with stochastic calculus ?

**Additional details for clarification**

The data we're getting is a time-series of 'black-box' forecasts for f(P), F(P),q(P) for each time t, where P is the instantaneous power and there will be around 100 forecasts for the interval I'd like to get the e(P) for. By 'Black-box' I mean that there will be a function I can call to evaluate f,F,q for P, but I don't know the underlying distribution.

The black-box functions are almost certainly interpolating output data from the model that produces the power forecasts, but we don't have access to that. I would guess that it **won't** be anything straightforward, since it comes from a chain of non-linear transformations. It's actually wind farm production forecasts: the wind speeds may be normally distributed, but multiple terrain and turbine transformations will change that.

**Further clarification** (I've edited the original text to remove confusing variable names in the energy distribution functions.)

The forecasts will be provided as follows:

The interval [t1,t2] that we need e, E and qe for is sub-divided into 100 (say) sub-intervals k=1...100. For each k we are given a distinct f(P), call them f_k(P). We need to calculate the energy distributions for the interval from this set of f_k(P).