04 Nov

This was originally written on Nov 3, 2013.

Converted from .tex using latex2wp.

Usually, we say a random variable ${X}$ follows a Normal(0,1) distribution, if its cumulative distribution can be expressed as:

$\displaystyle P\{X\leq t\}=\int_{-\infty}^{t}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}dx.$

Now we formalize this in a more measure-theoretic way, in correspondence to what we learned in the course, particularly, Read More

04 Nov

This was originally written on Nov 25, 2013.

Converted from .tex using latex2wp.

In this note, we explained why the conditional expectation of a random variable ${Y}$ given a ${\sigma}$-field ${\mathscr{G}}$ can be seen as “smoothed version of ${Y}$ over ${\mathscr{G}}$” (in Example 2), and we briefly related the definition of conditional expectation Read More