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I have two Gaussian distributions with mean and variance $(\mu_1,\sigma^2_1$) and $(\mu_2,\sigma^2_2)$. I then receive a series of values $x_1, x_2,...,x_n$ with mean and variance $(\mu_3,\sigma^2_3)$.

Assuming that there is equal chance that the the $x$'s were generated from the first or second Gaussians (and all of the $x$'s were generated from the same distribution), how can I determine - or even quantify - which of the two distributions is more likely to have been the source?

(In case I'm butchering terminology: I have a distribution for events of type A, and a distribution for events of type B, and I receive an unlabeled set of data - I want to determine whether it's more likely that it's data of events of type A or events of type B).

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If you believe ALL the new data come from exactly one of the two, you could try hypothesis testing. For individual data points, you might want to check out "Discriminant Analysis".

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You could try to apply a Parzen-Rosenblatt window on your $(x)$ distribution and compare it to your source distribution.

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