# Continuous random variable and the probability density function

Summary

Range

- The range of a continuous random variable is typically an interval.

Examples

- The length of a telephone call
- The volume of water in a bottle

Definition

- A (real-valued) random variable \(X\) is said to be continuous if there is a function \(f\) such that

\[P\left( a≤X≤b \right)=\int_{a}^{b}{ f\left( x \right)dx }\]

- for all real numbers \(a,b\) where \(a\) may be \(-∞\) and \(b\) may be \(∞\) .
- The function \(f\) is called the probability density function (pdf) of the random variable.
- The domain of \(f\) is always \(R\) and \(f\left( x \right)=0\) if \(x\) is not in the range of \(X\) .

Probability of inequalities

- A continuous random variable has infinite precision and \(P\left( X=x \right)=0\) for all \(x\) . Therefore, the following are always all equal:
- \(P\left( a≤X≤b \right)\)
- \(P\left( a<X<b \right)\)
- \(P\left( a≤X<b \right)\)
- \(P\left( a<X≤b \right)\)

Property of the pdf

- Since

\[P\left( -∞<X<∞ \right)=\int_{-∞}^{∞}{ f\left( x \right)dx }\]

- must be equal to one, the integral over all real numbers for any pdf must be one.