general formula to calculate the convolution of more than 2 probability distributions

## 2 Answers

You can define a convolution iteratively, as f\*g\*h = f\*(g\*h), etc. Or you could can extend the formula by summing over more index variables. So if you define

(g\*h)(n) = sum k=-∞..∞ g(k)h(n-k)

You could extend that as

(f\*g\*h)(n) = sum k=-∞..∞,j=-∞..∞ f(k)g(j)h(n-k-j)

You can continue in that fashion for more variables.

Or if you're asking how to compute it as a practical question, that depends. There's a lot of theory that might be relevant. Is this for a specific problem?
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The convolution formula is straight forward: the probability/probability density of a value is just the total probability/probability density of every way to sum up to that value.

So the formula is just p(X1+X2+...+Xn=k)=sum over all x1,...,xn such that x1+...+xn=k of p(X1=x1)p(X2=x2)....p(Xn=xn)

It's a bit inconvenient, but for faster computation you can use generating functions or Fourier transform.
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