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*.

## 3 comments:

I'm sure you condensed that to a huge degree... the one thing I can say, from what I understood ... is that you plan our model on reasonable expectations. Sometimes, it's linear; sometime', s, it's linear with multiple variables; sometimes it's exponential. But you ALWAYS figure out what the model is first, then you test the hypothesis...

I thought it pretty cool when I learned that one too. My difference was that I learned it first, then years later used it when teaching statistics at West Point. Most of the cadets weren't quite as fascinated as I was about the topic.

Stats are cool ... I think I've leaned Mary towards stats rather than BC senior year, Darren...

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