Andrews Curves

Posted on Thu 22 March 2018 in misc • 4 min read

#AcademicValentines

Posted on Thu 15 February 2018 in misc • 1 min read

Animated Heart in Python

Posted on Mon 08 February 2016 in misc • 3 min read

Expectation Maximisation in Python: Coin Toss Example

Posted on Fri 13 November 2015 in misc • 5 min read

Expectation Maximisation with Python : Coin Toss

This notebook implements the example, I consider a classic for understanding Expectation Maximisation.

See: http://www.nature.com/nbt/journal/v26/n8/full/nbt1406.html

Notations:

\begin{align*} \theta_A &= \text{Probability of a Heads showing up given the coin tossed is A}\\ \theta_B &= \text{Probability of a Heads showing up given the coin tossed is B}\\ \end{align*}

Entropy and Uniform Distribution

Posted on Sun 18 October 2015 in misc • 1 min read

To show \(\sum p_i\log(p_i) = p\)

Consider \(H = -\sum p_i log(p_i) = log(n)\)

Consider weighted AM-GM for \(p_i\):

$$ (\prod \frac{1}{p_i}^p_i)^\frac{1}{\sum p_i} \leq \sum \frac{1}{p_i} \times p_i = n $$

Take log both sides:

$$ \sum p_i \log(\frac{1}{p_i}) \leq \log(n …

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A frequent inequality

Posted on Wed 09 September 2015 in misc • 2 min read

$$ \frac{x-1}{x} \leq \ln(x) \leq x-1 \forall\ x>0 $$

Consider \(f(x)=\ln(x)-\frac{x-1}{x}\)

\(f'(x) = \frac{1}{x} - \frac{1}{x^2} = \frac{x-1}{x^2}\)

Now consider the following two cases:

Case A: \(0 < x \leq 1\) and Case B: \(1 < x < \infty …


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Functions of independent random variables are independent

Posted on Mon 01 June 2015 in misc • 1 min read

A textbook problem

Given \(X\), \(Y\) are two independent random variables, show that functions \(g(X)\) and \(h(Y)\) are independent too

Short Proof

Never took a course on measure theory, so avoiding that route:

Let.

\(R = g(X)\)

\(S = h(Y)\)

Also define,

\(A_r = \{x: g(X)\leq r …


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Bernoulli random varlable and UMVUE

Posted on Thu 21 May 2015 in misc • 1 min read

Problem

Let \(X_1,...,X_n\) random sample \(X\)~\(Bernoulli(p)\). For \(n\geq 4\) show that the product \(X_1X_2X_3X_4\) is a unbiased estimator for \(p^4\), and use this fact for find the best unbiased estimator of \(p^4\)

Solution

As posted on stackexchange

[ToDo: Add variance, prove \(E[\phi(T …


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Distribution of one random variable less than other

Posted on Mon 18 May 2015 in misc • 1 min read

Problem

\(P(Y<X)\) for any two independent random variables \(X,Y\)

Solution

We just follow the definition:

\(P(Y<X) = \int_{-\infty}^{\infty}f_X(x)dx \int_{-\infty}^{x}f_Y(y)dy\)

\(P(Y <X) = \int F_X(y)f_X(x)dx\)


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[Proof]Weighted AM-GM

Posted on Mon 18 May 2015 in misc • 1 min read

We make use of Jensen’s Inequality and the fact that \(log(x)\) is a concave function:

For concave f: \(f(\frac{\sum a_ix_i}{\sum a_i}) \geq \frac{\sum a_i f(x_i)}{\sum a_i}\)

\(f=log(X)\)

\(log(\frac{\sum a_ix_i}{\sum a_i}) \geq \frac{\sum a_i log(x_i)}{\sum …


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