Probability Density Function

Compact study note.

Summary

One probability density function describes how probability is distributed across a continuum. Probabilities are integrals of the density over intervals.^[1]

Prerequisites

• Continuous Random Variable

Notation and Assumptions

Density assumptions:

For measurable event set $C$ ,

Essential Result

Density values may exceed $1$ ; only integrated area is probability.

Small Example

For $X\sim \mathrm{Uniform}(0,0.5)$ , $f_X(x)=2$ on $(0,0.5)$ , but $P(0<X<0.25)=2(0.25)=0.5$ .

Common Mistakes

• Calling $f_X(3)$ the probability that $X=3$ .
• Forgetting to verify total area equals one.

Connections

• Cumulative Distribution Function
• Continuous Random Variable
• Continuous Uniform Distribution

References