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I’m reading a book that deals with the history of probability theory (Classic Problems of Probability by Prakash Gorroochurn). It is interesting to look back from a modern perspective and realize how many heated arguments could have been resolved in a few minutes if the participants would have had access to a computer. For example, Chapter 1 deals with Cardano and how he (understandably) confused probability with expectation. If you roll a die 3 times, the expectation value of the number of sixes is 0.

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I recently came across a surprising connection between the geometric mean and logarithms. At least, it was surprising to me, but isn’t that surprising once you see the derivation (a lot of things in math are this way). The connection is this. Take a set of observations \(y_1, y_2, \dots, y_n\). Assume that these are all positive numbers. Then \[ \langle \log(y) \rangle = \log(G).\] Here the angle brackets indicate the mean (expectation value), and \(G\) is the geometric mean of the observations.

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In a previous post, I said that an appropriate purgatory for physicists would be to have them read all of the words ever written on the interpretation of quantum mechanics. I have since noticed similar “purgatorial readings” in other fields. Here is a table: Field Purgatorial Reading Physics everything ever written on the interpretation of quantum mechanics Theology everything ever written on the Filioque History everything ever written on the Fall of Rome Statistics everything ever written on frequentist vs.

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In my last post, I discussed the geometric mean and how it relates to the more familiar arithmetic mean. I mentioned that the geometric mean is often useful for estimation in physics. Lawrence Weinstein, in his book Guesstimation 2.0, gives a mental algorithm for approximating the geometric mean. Given two numbers in scientific notation \[ a \times 10^x \quad \text{and}\quad b \times 10^y, \] where the coefficients \(a\) and \(b\) are both between 1 and 9.

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Given two positive numbers \(a\) and \(b\), the most well-known way to average them is to take the sum divided by two. This is often called the average or the mean of \(a\) and \(b\), but to distinguish it from other means it is called the arithmetic mean: \[ \text{arithmetic mean} = \frac{a + b}{2}. \] Another common, and useful, type of mean is the geometric mean. For two positive numbers \(a\) and \(b\), \[ \text{geometric mean} = \sqrt{ab}.

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