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Cumulative probability and probability density functions

The probability that an event will give values in the range x and $x + \Delta x$. Continuous distribution:


 \begin{displaymath}c.d.f.(x<=X)= \int_0^{X} p(x) dx
\end{displaymath} (3.1)

Discrete distribution:


 \begin{displaymath}c.d.f.(x<=X)= \sum_{i=1}^{N} p(x_i) dx
\end{displaymath} (3.2)

The expected value is


 \begin{displaymath}\langle x \rangle = \int_0^{\infty} x p(x) dx
\end{displaymath} (3.3)

Whereas the probability can be calculated from the ratio of the area of the p.d.f that corresponds to the condition that the event taking place to the area of the total curve, the probability can be read off directly from the cumulative distribution function (c.d.f.). The c.d.f starts with the value 0 and ends with 1. An example of a c.d.f is shown in Fig. 4.2b, in Chapter 4. The equivalent to a c.d.f for actual observations is the empirical cumulative function (e.d.f). Table 2 gives an overview of the four main types of distribution functions.


next up previous contents
Next: Normal/Gaussian distribution Up: What is probability distribution Previous: Distributions
David Stephenson
2000-09-02