If I understand your question correctly, you do not want the pdf of the sum of the random variables. The previous responses are all related to how to find the pdf of the sum of several random variables. Your question (it seems to me) is how to find the mixture of two or more random variables - see attached figure. Let me know if my interpretation of your question is correct.

Simple answer: p(sum(x_i))= p(x_1) * p(x_2) * p(x_3) * p(x_4) * ......, where the asterisk indicates the convolution operator.

 (I hope I did not misunderstand what you were asking for!)

Cheers!

Thank you George, I will try it, it seems reasonable!

Unfortunately it was not what I was looking for. Your suggestion calculates the pdf of X1 + X2 + X3 ... I was looking for the distribution if I put together all realisation of X1, X2, ...

Let t=0,1,2,3,...,n be the discrete time. Then X_0, X_1,..., X_n are random variables with their individual cdf  prob(X_1

If I understand your question correctly, you do not want the pdf of the sum of the random variables. The previous responses are all related to how to find the pdf of the sum of several random variables. Your question (it seems to me) is how to find the mixture of two or more random variables - see attached figure. Let me know if my interpretation of your question is correct.

I think the question needs to be reformulated more precisely.  Can you give an example of the application or problem you are trying to solve?  If you want to generate a mixture model from clustered sample data, perhams expectation-maximization is what you want.

Thanks for your answers! Noel Artiles-Leon is completely correct in what I was looking for!

If i understand your question, take only two variable X and Y not independent. If we klnow the pdf of X and the pdf of Y, can we deduce the jointe pdf of the couple (X,Y). The awnser is no because you need to have one  of the conditional distributions p(XIY) or p(YIX).

Kolmogorov extension theorem says that: If we know the partial distributions p(X_1, X_2, ... X_n) for any n,  under a consistant condition on pdfs then there is a unique pdf on the real sequences space wich is the pdf of the process (X_k)_{k\geq1}

I hope this helps you.

Thanks Brahim. Noel pointed me in the right direction. It's called Mixture Model or Mixture Distribution and is exactly what I was looking for. It's easier than I thought. I just have to add all components of the mixture distribution and weighten them. In my case all mixture components have the same weight, so I just have to divide the sum by the number of components.

So thanks for all your answers, this question is solved to me.

You are welcome! ... I am glad I was able to help.