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July 3, 2020

How to Pursue Mathematics after High School?

For Students who are passionate for Mathematics and want to pursue it for higher studies in India and abroad.

This is an interesting problem from ISI MSTAT PSB 2011 Problem 4 that tests the student's knowledge of how he visualizes the normal distribution in higher dimensions.

The Problem: ISI MSTAT PSB 2011 Problem 4

Suppose that \( X_1,X_2,... \) are independent and identically distributed \(d\) dimensional normal random vectors. Consider a fixed \( x_0 \in \mathbb{R}^d \) and for \(i=1,2,...,\) define \(D_i = \| X_i - x_0 \| \), the Euclidean distance between \( X_i \) and \(x_0\). Show that for every \( \epsilon > 0 \), \(P[\min_{1 \le i \le n} D_i > \epsilon] \rightarrow 0 \) as \( n \rightarrow \infty \)

Prerequisites:

  1. Finding the distribution of the minimum order statistic
  2. Multivariate Gaussian properties

Solution:

First of all, see that \( P(\min_{1 \le i \le n} D_i > \epsilon)=P(D_i > \epsilon)^n \) (Verify yourself!)

But, apparently we are more interested in the event \( \{D_i < \epsilon \} \).

Let me elaborate why this makes sense!

Let \( \phi \) denote the \( d \) dimensional Gaussian density, and let \( B(x_0, \epsilon) \) be the Euclidean ball around \( x_0 \) of radius \( \epsilon \) . Note that \( \{D_i < \epsilon\} \) is the event that the gaussian \( X_i \) will land in this Euclidean ball.

So, if we can show that this event has positive probability for any given $x_0, \epsilon$ pair, we will be done, since then in the limit, we will be exponentiating a number strictly less than 1 by a quantity that is growing larger and larger.

In particular, we have that : \( P(D_i < \epsilon)= \int_{B(x_0, \epsilon)} \phi(x) dx \geq |B(x_0, \epsilon)| \inf_{x \in B(x_0, \epsilon)} \phi(x) \) , and we know that by rotational symmetry and as Gaussians decay as we move away from the centre, this infimum exists and is given by \( \phi(x_0 + \epsilon \frac{x_0}{||x_0||}) \) . (To see that this is indeed a lower bound, note that \( B(x_0, \epsilon) \subset B(0, \epsilon + ||x_0||) \).

So, basically what we have shown here is that exists a \( \delta > 0 \) such that \( P(D_i < \epsilon )>\delta \).

As, \( \delta \) is a lower bound of a probability , hence it a fraction strictly below 1.

Thus, we have \( \lim_{n \rightarrow \infty} P(D_i > \epsilon)^n \leq \lim_{n \rightarrow \infty} (1-\delta)^n = 0 \).

Hence we are done.

Food for thought:

There is a fantastic amount of statistical literature on the equi-density contours of a multivariate Gaussian distribution .

Try to visualize them for non singular and a singular Gaussian distribution separately. They are covered extensively in the books of Kotz and Anderson. Do give it a read!

Some Useful Problems:

What to do to shape your Career in Mathematics after 12th? 

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

  • What are some of the best colleges for Mathematics that you can aim to apply for after high school?
  • How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
  • What are the best universities for MS, MMath, and Ph.D. Programs in India?
  • What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
  • How can you pursue a Ph.D. in Mathematics outside India?
  • What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Want to Explore Advanced Mathematics at Cheenta?

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2 comments on “ISI MSTAT PSB 2011 Problem 4 | Digging deep into Multivariate Normal”

  1. Anderson has several books and two of them are multivariate and elliptical countours. Would you like to say which one do you mention?

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