statistics - Integrating Power pdf to get energy pdf? -

i'm trying work out how solve seems simple problem, can't convince myself of correct method.

i have time-series data represents pdf of power output (p), varying on time, cdf , quantile functions - f(p,t), f(p,t) , q(p,t). need find pdf, cdf , quantile function energy in given time interval [t1,t2] data - e(), e(), , qe().

clearly energy integral of power on [t1,t2], how best calculate e, e , qe ?

my best guess since q(p,t) power, should generate qe integrating q on time interval, , calculate other distributions that.

is simple that, or need grips stochastic calculus ?

additional details clarification

the data we're getting time-series of 'black-box' forecasts f(p), f(p),q(p) each time t, p instantaneous power , there around 100 forecasts interval i'd e(p) for. 'black-box' mean there function can call evaluate f,f,q p, don't know underlying distribution.

the black-box functions interpolating output data model produces power forecasts, don't have access that. guess won't straightforward, since comes chain of non-linear transformations. it's wind farm production forecasts: wind speeds may distributed, multiple terrain , turbine transformations change that.

further clarification (i've edited original text remove confusing variable names in energy distribution functions.)

the forecasts provided follows:

the interval [t1,t2] need e, e , qe sub-divided 100 (say) sub-intervals k=1...100. each k given distinct f(p), call them f_k(p). need calculate energy distributions interval set of f_k(p).

thanks clarification. can tell, don't have enough information solve problem properly. specifically, need have estimate of dependence of power 1 time step next. longer time step, less dependence; if steps long enough, power might approximately independent 1 step next, news because simplify analysis quite bit. so, how long time steps? hour? minute? day?

if time steps long enough independent, distribution of energy distribution of 100 variables, distributed central limit theorem. it's easy work out mean , variance of total energy in case.

otherwise, distribution more complicated result. guess variance estimated independent-steps approach big -- actual variance less, believe.

from say, don't have information temporal dependence. maybe can find or derive other source or sources estimate autocorrelation function -- wouldn't surprised if question has been studied wind power. wouldn't surprised if general version of problem has been studied -- perhaps can search "distribution of sum of autocorrelated variables." might interest in question on stats.stackexchange.com.