Cumulative Distribution Function
================================

The cumulative distribution function brings the discrete and
continuous RV's toegether. For a RV $X$ the cumulative distribution
function (often called the distribution function) is defined as:

.. math::
   F_X(x) = \P(X\leq x)

Note that $x\in\setR$ even in case $X$ is a discrete RV. We have:

.. math::
   F_X(x) = \begin{cases}
   \sum_{k=-\infty}^{\lfloor x\rfloor} p_X(k) &\text{Discrete $X$}\\
   \int_{-\infty}^{x} f_X(y)\, dy &\text{Continuous $X$}
   \end{cases}


Below a probability mass function $p_X$ is plotted and the
corresponding $F_X$.

.. ipython:: python
   :suppress:
      
   from matplotlib.pylab import *

   pX = array([0,0,0,0,1,2,4,5,7,6,5,3,1,0,0,0])
   pX = pX / sum(pX)
   x = arange(len(pX))-7
   cpX = cumsum(pX)
   subplot(211)
   title(r"Probability Mass Function")
   stem(x,pX)
   subplot(212)
   title(r"Cumulative Distribution Function")
   step(x, cpX, where='post')
   @savefig discrete_cdf.png
   show()
   
And a plot of a probability density function and its corresponding
cumulative distribution function.

.. ipython:: python
   :suppress:

   #
   # I am cheating by calculating things for a sampled function...
   # ONLY LOOK AT THE PLOTS...
   #
   from matplotlib.pylab import *

   x = linspace(-8, 8, 1000)
   pX = piecewise(x, [x<-4, logical_and(x>=-4, x<4), x>=4],
                  [0, lambda x: 16-x**2, 0])
   pX = pX / sum(pX)
   cpX = cumsum(pX)

   subplot(211)
   title(r"Probability Density Function")
   plot(x,pX)
   subplot(212)
   title(r"Cumulative Distribution Function")
   step(x, cpX, where='post')
   @savefig continuous_cdf.png
   show()

  
The cumulative distribution function follows from the probability
density function by integration. We can go the other way as well:

.. math::
   f_X = \frac{d}{dx} F_X

With some mathematical leniency we could say that this also holds for
a discrete random variable.