# 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.