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# 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 a very beautiful sample problem from ISI MStat PSB 2012 Problem 9, It's about restricted MLEs, how restricted MLEs are different from the unrestricted ones, if you miss delicacies you may miss the differences too . Try it! But be careful.

## Problem- ISI MStat PSB 2012 Problem 9

Suppose $X_1$ and $X_2$ are i.i.d. Bernoulli random variables with parameter $p$ where it us known that $\frac{1}{3} \le p \le \frac{2}{3}$. Find the maximum likelihood estimator $\hat{p}$ of $p$ based on $X_1$ and $X_2$.

### Prerequisites

Bernoulli trials

Restricted Maximum Likelihood Estimators

Real Analysis

## Solution :

This problem seems quite simple and it is simple, if and only if one observes subtle details. Lets think about the unrestricted MLE of $p$,

Let the unrestricted MLE of $p$ (i.e. when $0\le p \le 1$ )based on $X_1$ and $X_2$ be $p_{MLE}$, and $p_{MLE}=\frac{X_1+X_2}{2}$ (How ??)

Now lets see the contradictions which may occur if we don't modify $p_{MLE}$ to $\hat{p}$ (as it is been asked).

See, that when if our sample comes such that $X_1=X_2=0$ or $X_1=X_2=1$, then $p_{MLE}$ will be 0 and 1 respectively, where $p$, the actual parameter neither takes the value 1 or 0 !! So, $p_{MLE}$ needs serious improvement !

To, modify the $p_{MLE}$, lets observe the log-likelihood function of Bernoulli based in two samples.

$\log L(p|x_1,x_2)=(x_1+x_2)\log p +(2-x_1-x_2)\log (1-p)$

Now, make two observations, when $X_1=X_2=0$ (.i.e. $p_{MLE}=0$), then $\log L(p|x_1,x_2)=2\log (1-p)$, see that $\log L(p|x_1,x_2)$ decreases as p increase, hence under the given condition, log_likelihood will be maximum when p is least, .i.e. $\hat{p}=\frac{1}{3}$.

Similarly, when $p_{MLE}=1$ (i.e.when $X_1=X_2=1$), then for the log-likelihood function to be maximum, p has to be maximum, i.e. $\hat{p}=\frac{2}{3}$.

So, to modify $p_{MLE}$ to $\hat{p}$, we have to develop a linear relationship between $p_{MLE}$ and $\hat{p}$. (Linear because, the relationship between $p$ and $p_{MLE}$ is linear. ). So, $\hat{p}$ and $p_{MLE}$ is on the line that is joining the points $(0,\frac{1}{3})$ ( when $p_{MLE}= 0$ then $\hat{p}=\frac{1}{3}$) and $(1,\frac{2}{3})$. Hence the line is,

$\frac{\hat{p}-\frac{1}{3}}{p_{MLE}-0}=\frac{\frac{2}{3}-\frac{1}{3}}{1-0}$

$\hat{p}=\frac{2-X_1-X_2}{6}$. is the required restricted MLE.

Hence the solution concludes.

## Food For Thought

Can You find out the conditions for which the Maximum Likelihood Estimators are also unbiased estimators of the parameter. For which distributions do you think this conditions holds true. Are the also Minimum Variance Unbiased Estimators !!

Can you give some examples when the MLEs are not unbiased ?Even If they are not unbiased are the Sufficient ??

## 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?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

To Explore and Experience Advanced Mathematics at Cheenta

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