p^2-q and q^2-p prime

p and q are 2 prime numbers. p^2 - q and p - q^2 are also prime. If you divide p^2 - q by a composite number n, where n < p, you’ll get a remainder of 14. If you divide p - q^2 + 14 by the same number, what will you get as the remainder?” – Akash

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Statistical odds and ends

Between school and family duties, I’ve been finding it hard to find any time to indulge in olympiad math blogging 😦 At the same time, I’ve missed the feeling of typing up stuff that I find interesting and sharing it with others.

To that end, I just started a new blog Statistical Odds and Ends! The idea for this began when I found myself spending a lot of time googling relatively simple things in the course of my studies and research. For example:

  • Why does the ridge regression solution exist and why is it unique?
  • What is the formula for the matrix P such that the projection of the vector v onto the column space of a matrix A is Px?
  • Can I switch supremums and expectations and still have equality? If not, can I get an inequality instead?
  • How can I derive the bias-variance decomposition?

I was often googling for the same things over and over again, and trying to re-understand what others were writing.

Hence the idea of Statistical Odds and Ends. The blog will be a place for me to pen down my understanding of these statistical tidbits, and to share it with others. Hopefully some of the material there will be of interest to you! If the content is relevant to this audience, I will cross-post over on this blog too.

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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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