Continuous Random Variable

Compact study note.

Summary

One continuous random variable is commonly represented by a probability density function. Probabilities come from areas under the density, not from density values at points.^[1]

Prerequisites

• Random Variable
• Probability Density Function

Notation and Assumptions

One continuous random variable with density $f_X$ has $F_X(x)={\int }_{-\mathrm{\infty }}^x{f}_X(u)du$ and $f_X(u)\ge 0$ with total integral $1$ .

Essential Result

For interval endpoints $l<u$ ,

Also $P(X=c)=0$ for every single point $c$ .

Small Example

If $X\sim \mathrm{Uniform}(0,1)$ , then $P(0.2<X<0.5)={\int }_{0.2}^{0.5}1dx=0.3$ .

Common Mistakes

• Interpreting $f_X(x)$ itself as point probability.
• Expecting endpoint inclusions to matter for continuous distributions.

Connections

• Probability Density Function
• Cumulative Distribution Function
• Normal Distribution

References