Probability for Machine Learning


By Prof. Seungchul Lee
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH

Table of Contents

1. Random Variable (= r.v.)

1.1. Definition


  • (Rough) Definition: Variable with a probability
  • Probability that $x=a$


$$\triangleq \; P_X(x=a)\;= \; P(x=a) \;\implies\; \begin{cases} \text{1)} \; P(x=a) \geq 0 \\ \text{2)} \; \sum\limits_{\text{all}} P(x)=1 \end{cases}$$


  • $\begin{cases} \text{continuous r.v.} \qquad \text{if} \;x \;\text{is continuous}\\ \text{discrete r.v.} \quad \qquad \, \text{if} \;x \;\text{is discrete} \end{cases}$

Example

  • $x$: die outcome


$$ P(x=1)=P(x=2)= \;\dotsb \;= P(x=6) = \frac{1}{6} $$

  • Question
$$ \begin{align*} y &= x_1 + x_2: \;\;\; \text{ sum of two dice} \\\\ P_Y(y=5) &= \text{?} \end{align*} $$