# Multiplying two probability distributions represented by particles

Hello,

I have two probability distributions p1(x) and p2(x) given by (x_1i, w_1i) and (x_2i, w_2i) respectively, i.e. they are both represented by sets of particles. I need to create the pdf p(x) = p1(x) . p2(x), their product. I read that for functions supported on integers (these would be like the delta functions of the particles), multiplying is the same as convolution. But I still can't figure out how to create a new pdf represented by (x_i, w_i) from these two sets. Can somebody give me some ideas?

Thanks

• Usually when you multiply probabilities, they are probabilities for different events, not for the same event. I wonder if you really want p1(x)p2(x) or actually p1(x)p2(y). Multiplying delta functions depending on different arguments is no problem. I assume when you say p(x) is represented by $(x_i,w_i)$, you mean $x=x_i$ with probability $w_i$. With this interpretation $p1(x)p2(y)$ is represented by $(x_iy_j,w_{1i}w_{2j})$. – Michael Renardy May 23 '11 at 16:00
• NBP has quite a definite algorithm. I do not quite understand which part could cause such a confusion whether to use a product of PDFs or a convolution. Anyway, if it helps, a product of PDFs is the same as a convolution of their characteristic functions. I'm just guessing that that might be what you were referring to. If you just multiply values of probabilities on the same outcome, do not forget to normalize the result. If you multiply outcomes too, be aware that some pairs of outcomes might give the same product. – Tunococ Jun 20 '11 at 10:28