2.5.2. Continuous Random Variables

If we measure the length of a random person then the length \(X\) is a continous random variable, in principle the length can be any value \(x\in\setR\). The peculiarity of continuous random variables is that \(\P(X=x)=0\) for all \(x\in\setR\). What else could the probability of finding someone with length \(180+\pi\) cm be? Therefore a probility mass function is non sensical for continuous random variables.

Instead we define the probability density function of a continuous RV as the function \(f_X: x\in\setR \mapsto f_X(x)\in\setR\) such that:

\[\P(a\leq X \leq b) = \int_a^b f_X(x)\,dx\]

Note that:

1. The probability for \(-\infty\leq x \leq\infty\) should be one (the entire real line of course is the universe for this RV) and therefore:

   \[\int_{-\infty}^{\infty} f_X(x)\,dx = 1\]

2. A probability density is not a probability, we have to integrate the density over a subset in \(\setR\) to get a probability.

3. We have

   \[f_X(x)\geq 0\]

   but be aware that densities can be larger then 1.