Here's a really excellent weather forecast graphic that the NOAA website has been producing for ages. I haven't found anyone else who makes one as useful. To find yours, go to weather.gov (easier to remember than to bookmark), put in your zip code, and search the forecast page for the word "hourly."

I talk frequently about my wise who says "I believe weather reports, like 'it rained.'" But if I want to dissect how much to believe a forecast, this is the best one. Let's break it down.

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I am reminded today by a video by "Mathemaniac" about the James-Stein estimator.

Suppose I'm trying to estimate a number $n$ of independent parameters simultaneously, by taking a sample from each one-dimensional normal distribution with unknown means $\mu_n$ and unit standard deviations $\sigma_n=1$. The naïve estimator is to use each sample $x_n$ as an estimate $\hat\mu_n$ of the mean. However, if my number of parameters is large enough, the zero-biased estimator

$$ \left(\begin{array}{c} \hat \mu_1 \\ \vdots \\ \hat \mu_n \end{array}\right) = \left( 1-\frac{n-2}{x_1^2 + \cdots + x_n^2} \right) \left(\begin{array}{c} x_1 \\ \vdots \\ x_n \end{array}\right) $$

actually produces a smaller mean-squared error on the ensemble as a whole.

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Consider the following decimal expansions:

$$\begin{aligned} \textstyle \frac{1}{ 1 } &= 1.0 & \textstyle \frac{1}{ 11 } &= 0.[09] & % 0.09090909090909091 \\ \textstyle \frac{1}{ 21 } &= 0.[04761\,9] \\ % 0.047619047619047616 \\ % \textstyle \frac{1}{ 2 } &= 0.5 & \textstyle \frac{1}{ 12 } &= 0.08[3] & % 0.08333333333333333 \\ \textstyle \frac{1}{ 22 } &= 0.0[45] \\ % 0.045454545454545456 \\ % \textstyle \frac{1}{ 3 } &= 0.[3] & % 0.3333333333333333 \\ \textstyle \frac{1}{ 13 } &= 0.[07692\,3] & % 0.07692307692307693 \\ \textstyle \frac{1}{ 23 } &= 0.[04347\,82608\,69565\,21739\,13] \\ % 0.043478260869565216 \\ % \textstyle \frac{1}{ 4 } &= 0.25 & \textstyle \frac{1}{ 14 } &= 0.0[71428\,5] & % 0.07142857142857142 \\ \textstyle \frac{1}{ 24 } &= 0.041[6] \\ % 0.041666666666666664 \\ % \textstyle \frac{1}{ 5 } &= 0.2 & \textstyle \frac{1}{ 15 } &= 0.0[6] & % 0.06666666666666667 \\ \textstyle \frac{1}{ 25 } &= 0.04 \\ % \textstyle \frac{1}{ 6 } &= 0.1[6] & % 0.16666666666666666 \\ \textstyle \frac{1}{ 16 } &= 0.0625 & \textstyle \frac{1}{ 26 } &= 0.0[38461\,5] \\ % 0.038461538461538464 \\ % \textstyle \frac{1}{ 7 } &= 0.[14285\,7] & % 0.14285714285714285 \\ \textstyle \frac{1}{ 17 } &= 0.[05882\,35294\,11764\,7] & % 0.058823529411764705 \\ \textstyle \frac{1}{ 27 } &= 0.[037] \\ % 0.037037037037037035 \\ % \textstyle \frac{1}{ 8 } &= 0.125 & \textstyle \frac{1}{ 18 } &= 0.0[5] & % 0.05555555555555555 \\ \textstyle \frac{1}{ 28 } &= 0.035[71428\,5] \\ % 0.03571428571428571 \\ % \textstyle \frac{1}{ 9 } &= 0.[1] & % 0.1111111111111111 \\ \textstyle \frac{1}{ 19 } &= 0.[05263\,15789\,47368\,421] & % 0.05263157894736842 \\ \textstyle \frac{1}{ 29 } &= 0.[03448\,27586\,20689\,65517\,24137\,931] \\ % 0.034482758620689655 \\ % \textstyle \frac{1}{ 10 } &= 0.1 & \textstyle \frac{1}{ 20 } &= 0.05 & \textstyle \frac{1}{ 30 } &= 0.0[3] \\ % 0.03333333333333333 \\ \end{aligned}$$

Each of these either terminates, like $\frac18 = 0.125$, or repeats, like $\frac{1}{27} = 0.[037]$. What determines the length of these repeating sequences?

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