The current course I'm taking for my master's program, Mathematical Statistics, is difficult but fun. We're learning how to estimate function parameters based on data. An easy example would be if you believe that your data are distributed in accordance with a normal distribution, and based on the data you try to determine mu and sigma (the mean and standard deviation). Normal distributions are pretty easy; what would you do if your data is explained by an exponential distribution (parameter is lambda), or a gamma distribution (two parameters, alpha and lambda)?
I'm going to get a bit "inside baseball" here, but we're talking about sufficient statistics--instead of looking at an entire set of data, what summary statistics of that data would suffice to allow us to estimate the unknown parameters? It's not always mean and standard dev! If our function is part of the "exponential family" of functions, we can rewrite the function in such a certain way that the sufficient statistics become apparent.
A few months ago my stats students learned about the binomial distribution--a distribution that allows you to determine, for example, the likelihood of getting 7 heads if you flip a fair coin 10 times (it's a little under 12%). The formula is f(x|p)=nCx*p^x*(1-p)^(n-x).
Recently my pre-calculus students have been reviewing logarithm operations and functions.
So in my Mathematical Statistics lesson tonight we were determining the sufficient statistics for the binomial distribution. The way the distribution is written above, though, it doesn't (at first glance) appear to be of the exponential family. However, if we employ a little advanced algebra, we find that it is of the exponential family and can be rewritten in a way that reveals the sufficient statistic.
The algebraic manipulations? e^(ln x)=x and ln a^b=b ln a, just the material I've been going over with my pre-calculus students. Using those facts I can rewrite p^x as e^(x ln p) and (1-p)^(n-x) as e^(n-x)ln(1-p). I'll skip the rest of the work, just noting that I think it's cool when I can tie what I'm learning directly to what I'm teaching. It's the reason I'm pursuing this particular degree instead of a generic Master's Degree in Education, because this degree is actually going to make me a better math teacher.