Stats Joke

From page 68 of Simon Singh’s The Simpsons and their Mathematical Secrets:

While heading to a conference on board a train, three statisticians meet three biologists. The biologists complain about the cost of the train fare, but the statisticians reveal a cost-saving trick. As soon as they hear the inspector’s voice, the statisticians squeeze into the toilet. The inspector knocks on the toilet door and shouts: “Tickets, please!” The statisticians pass a single ticket under the door, and the inspector stamps it and returns it. The biologists are impressed. Two days later, on the return train, the biologists showed the statisticians that they have bought only one ticket, but the statisticians reply: “Well, we have no ticket at all.” Before they can ask any questions, the inspector’s voice is heard in the distance. This time the biologists bundle into the toilet. One of the statisticians secretly follows them, knocks on the toilet door and asks: “Tickets please!” The biologists slip the ticket under the door. The statistician takes the ticket, dashes into a another toilet with his colleagues, and waits for the real inspector. The moral of the story is simple: “Don’t use a statistical technique that you don’t understand.”

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[Soln] Central Limit Theorem: Strange Result!

For n \in \mathbb{N}, define the random variable

X_n = \begin{cases} \pm 1 &\text{each with probability } \frac{1}{2}\left( 1 - \frac{1}{n^2} \right), \\ \pm n^2 &\text{with probability } \frac{1}{2n^2}. \end{cases}

Let S_n = \displaystyle\sum_{k = 1}^n X_k. Prove that as n \rightarrow \infty,

a) the distribution of \displaystyle\frac{S_n}{\sqrt{n}} converges to \mathcal{N}(0, a) for some real number a \neq 2,

b) but \text{Var} \displaystyle\frac{S_n}{\sqrt{n}} converges to 2.

(Credits: I learnt of this problem from Persi Diaconis in my probability class.)

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[Hints] Central Limit Theorem: Strange Result!

Click for hints for Central Limit Theorem: Strange Result!

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Central Limit Theorem: Strange Result!

For n \in \mathbb{N}, define the random variable

X_n = \begin{cases} \pm 1 &\text{each with probability } \frac{1}{2}\left( 1 - \frac{1}{n^2} \right), \\ \pm n^2 &\text{with probability } \frac{1}{2n^2}. \end{cases}

Let S_n = \displaystyle\sum_{k = 1}^n X_k. Prove that as n \rightarrow \infty,

a) the distribution of \displaystyle\frac{S_n}{\sqrt{n}} converges to \mathcal{N}(0, a) for some real number a \neq 2,

b) but \text{Var} \displaystyle\frac{S_n}{\sqrt{n}} converges to 2.

(Credits: I learnt of this problem from Persi Diaconis in my probability class.)

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[Soln] 2016 Putnam Problem B1

2016 Putnam B1. Let x_0, x_1, x_2, \dots be the sequence such that x_0 = 1 and for n \geq 0, x_{n+1} = \ln(e^{x_n} - x_n) (as usual, the function  \ln is the natural logarithm.

Show that the infinite series x_0 + x_1 + x_2 + \dots converges and find its sum.

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2016 Putnam Problem B1

2016 Putnam B1. Let x_0, x_1, x_2, \dots be the sequence such that x_0 = 1 and for n \geq 0, x_{n+1} = \ln(e^{x_n} - x_n) (as usual, the function  \ln is the natural logarithm.

Show that the infinite series x_0 + x_1 + x_2 + \dots converges and find its sum.

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Data on IMO results

Following the recent IMO 2016, I have been meaning to do some analysis on IMO results. Unfortunately I have not had time to do so…

In the meantime, I thought I’d share the data I’ve scraped so far so that others who have the time and interest might have a go at analysing the data. Data is available at my Github repo.

All data was scraped from imo-official.org; the scripts I used to scrape them are in the ETL folder of the same repo. The data generally looks clean except some minor issues for “Contestant” (i.e. names of contestants). For example, my name is written in one order for 2003 and 2005 but in a different way for 2004. I have no idea how widespread this issue is, although a cursory glance at contestants in my country suggest the issue is a minor one.

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