Posts on n4735 n0735

Probability Part II

Sun, 11 Oct 2026 00:00:00 +0000

$$ \gdef\dx{\mathrm{d}x} \gdef\dm{\mathrm{d}m} \gdef\dr{\mathrm{d}r} $$

$\gdef\dx{\mathrm{d}x}$ $\gdef\dm{\mathrm{d}m}$ $\gdef\dr{\mathrm{d}r}$

Intro

Last time, in Part I, we covered the basic concepts leading up to random variables and their distributions. In this post, things will start to get cracking as we build up to the law of the unconscious statistician, the law of large numbers, and Monte Carlo integration. Along the way, we’ll introduce the following terms:

Term	Notation
Expected Value of $X$ under distribution $p$	$\mathbb{E}_p[X]$
Variance of $X$ under distribution $p$	$\mathbb{V}_p[X]$

Expected Value

The expected value $\mathbb{E}_p[X]$ of a random variable $X$ with distribution $p$ is the probability-weighted average of its possible values. Intuitively, it represents the average value we would observe over many independent repetitions of the same experiment.

Probability Part I

Sun, 27 Sep 2026 00:00:00 +0000

Intro

Computer graphics often involves estimating complicated lighting integrals using Monte Carlo integration. This requires a background in random numbers and probability. By the end of this post you should understand the basics of probability, including the following terms and notations:

Term	Notation
Outcome	$s$
Set of Outcomes	$S$
Event	$A$
Event Space	$\mathcal{S}$
Probability Measure	$\mathbb{P}$
Sample Space	$(S, \mathcal{S})$
Probability Space	$(S, \mathcal{S}, \mathbb{P})$
Random Variable	$X$
Probability Distribution of $X$	$\mathbb{P}_X$
Probability Mass Function (PMF)	$p_X$
Cumulative Distribution Function (CDF)	$F_X$
Probability Density Function (PDF)	$f_X$

Example: Rolling a die

Let’s roll a fair six-sided die.

Tangent Plane

Mon, 01 Nov 2021 00:00:00 +0000

Explicit vs Implicit Surface

Let’s say that you have a surface given by $z=f(x,y)$.

You want to find the tangent plane to that surface at a given point $p_0 = (x_0, y_0, z_0)$. There are two tangent plane equations to choose from.

The explicit equation

$$ z = f(x_0, y_0) + f_x(x_0, y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0) $$

Or the implicit equation

$$ 0 = \nabla f \cdot (p - p_0 ) $$ where $p = (x, y, z)$