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:

.. math::
   \P(a\leq X \leq b) = \int_a^b f_X(x)\,dx

Note that:

#. 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:

   .. math::
      \int_{-\infty}^{\infty} f_X(x)\,dx = 1


#. A probability *density* is not a probability, we have to integrate
   the density over a subset in $\setR$ to get a probability.

#. We have

   .. math::
      f_X(x)\geq 0

   but be aware that densities can be larger then 1.