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I have two discrete dependent random variables $X,Y$, where both $X$ and $Y$ can take values either $0$ or $1$. Furthermore, I know their joint distribution $f_{X,Y}(X,Y)$.

Now let's say I have an iid sequence $(X_1, Y_1) \dots, (X_K,Y_K)$, with $K >0$ known. I'm interested in computationally obtaining the probability distribution of $$f_{\sum_{k =1}^K X_k,\sum_{k =1}^K Y_k} \left(\sum_{k =1}^K X_k, \sum_{k =1}^K Y_k \right).$$

Does anyone know if I can do this via convolutions/Fast Fourier transforms? And if so how this can be done?

I'd ideally like to generate a $K+1$ by $K+1$ matrix whose $i,j$th entry is $\mathbb{P}(\sum_{k =1}^K X_k = i-1, \sum_{k =1}^K Y_k = j-1)$.

Thanks!

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  • $\begingroup$ Convolution generalizes naturally to multivariate arrays and the FFT method similarly generalizes. $\endgroup$ – whuber Nov 14 '19 at 23:50

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