Posts

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

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. ...

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 $p = (x, y, z)$ ...