## 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.”

## [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.)

## 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.)

## [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.

## 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.