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I am trying to distinguish between two letters (A and C) using the difference of sums of black pixels in their left and right half of the image. I assume they normal distribution of the class probability density functions p(x|k), where k={A, C}. I have to find mean and sigma (normal distribution is defined by them) using MLE. I have training data for estimations and test data to verify my estimates.

I know hot to use MLE to estimate only one parametre, but how to estimate both at once? Lets assume I know the a priori probabilities of P(A) and P(C).

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For the normal distribution, the MLE gives exactly the same result as reading the mean and the variance from the sample directly. So, just figure out the parameters for each letter separately in the usual way. After that, all you need to do is to multiply the a priori probability of each letter by the density of the corresponding Gaussian at the observed difference and take the letter with the largest product (that's the standard Bayes).

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