Combined distribution function from multiple identical distributions and from two different distributions

Hi everyone, i tried to find the answer for my question in this forum and on Google, but unfortunately i was unsuccessful. I hope one of you can help me.

My first question is how to generate a general cdf and pdf for describing a random variable that is a combination of multiple draws from the same distribution function. So for example the distribution function for a weeks demand, when knowing the demand parameters for a days demand.

Dweek = sum(D1 + ... +D5) where D1 to D5 follow the same distribution function.

(I need a general function, as my program should be capable of handling different distribution functions)

And the second question is how to generate a general cdf and pdf for describing a random variable that describes the difference of two different distributed random variables.

The final application for the problem is to get the probability of cumulated deliveries exceeding cumulative demands so:

P(sum(Xi)> sum(Yi)) with i=1:5 as control variable

i already found out that it can be rearanged to P(sum(Xi - sum(Yi) = 0)

I hope you can understand my problem, as it is a bit hard to formulate and it would be great if somebody could help me with it.

Answer 1

[Matt J]

The question looks similar to this one.

[Alex]

Thanks for you answer, but unfortunately i need to solve it analytically as it is part of a model that depends on the exactness of this combined distribution function, sampling would therefore be not exact enough, but thanks for your help, i already considered sampling and i will try that if i cant get the exact solution working.

[Matt J]

I doubt you will find a closed form result for arbitrarily distributed Di.