Continuous Random Variables

calendar_today March 31, 2026 article 91 words access_time 1 min replay Modified March 31, 2026

Table of Contents

Continuous Range

A random variable can have a continuous range
Typical ranges are intervals of $\mathbb R$
Common examples
- arrival time with range $[0,\infty)$
- voltage with range $(-\infty,\infty)$
- phase with range $[0,2\pi)$

Proposition 14.1 (Point Probabilities for Continuous RV).

If $X$ is continuous then for any real number $x$

$$ P[X=x]=0 $$

For continuous RVs, probability is assigned to intervals not points
Small intervals can have small probability
Interval endpoints do not matter for probabilities since single points have probability zero

Corollary 14.2 (Endpoint Irrelevance).

If $X$ is continuous and $a $$ P[a

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