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Given a sample, one can usually find the best fitting normal distribution by matching the mean and variance. What's the correct way to fit a normal distribution to data when the parameters aren't constant, but rather functions of another field?

Specifically, I have observations at various times, and I expect the difference between two observations at t1 and t2 to be normally distributed with mean u(t2-t1) and variance s2(t2-t1)+err2. How can I find the values of u, s, and err that best fit the data?

I've considered transforming my data to make the Normal parameters constant, and it's simple enough to divide by (t2-t1) so that the mean will be just u, but I haven't figured out how I could transform the data to make the variance constant since there are 2 separate variables to estimate for the variance.

Bonus question: Can I match every observation with every other to get n*(n-1) pairs for estimating the parameters or does that "double-count" my data and I need to only match each ti with ti+1?

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