mrkb80 said:
Point taken. I'm going down this path now:
F_{A+B+C} = P(A+B+C \le x)
= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{x-b-c} f_A(a) f_B(b) f_C(c) da db dc
= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} F_A(x-b-c) f_B(b) f_C(c) db dc
Not sure where to go next...
I doubt that the task is 'doable' analytically. Probably you need to resort to a numerical method.
Probably the easiest way is through successive convolution: if X = B+C, then its density is
f_X(x) = \int_0^x f_B(y) f_C(x-y) \, dy.
Then the density of S = A+B+C = A + X is
f_S(x) = \int_0^x f_A(y) f_X(x-y) \, dy
Note that the integrations only go from 0 to x, not from -∞ to +∞; this is because the random variables are all ≥ 0.
However, at this point I think you are stuck: you probably need to compute and store the values of ##f_X(x)## on a grid of x-values, or maybe come up with a convenient approximate formula for it, because in some cases at least we can
prove that there is no finite formula for ##f_X## in terms of elementary functions. We can establish this by example: suppose B and C are iid Weibull with parameters k = 2 and λ = 1, so that the density of B (and C) is
f_B(x) = f_C(x) = 2x e^{-x^2}.
We can actually do the integral to get ##f_X##:
f_X(x) = x e^{-x^2} + \sqrt{\frac{\pi}{2}}\,(x^2-1)\, e^{ -x^2 /2}\, \text{erf}(x/\sqrt{2}).
It is impossible to give a finite, closed-form formula for this in terms of elementary functions, because if we could do it, we would have a finite, closed-form expression for 'erf', and that has been shown to be impossible. Of course, you can give non-finite expressions, such as infinite series and the like.
Anyway, that special case proves to be un-doable in simple terms, so the general case must also be un-doable.
All I can suggest is some type of numerical approximation when actual numerical inputs are specified. However, quantities like the mean and variance of S = A+B+C are easily obtained using standard result about moments of sums of independent random variables.
