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PDF discretisation

A probability density function (PDF) represents a probability distribution in terms of integrals. The probability of an interval $ [a,b]$ for a pdf $ f(x)$ is given by the integral

$\displaystyle \int_a^b f(x) dx
$

for any two number $ a$ and $ b$, $ a < b$. To fulfill this property, $ f(x)$ has to be a non-negative Lebesque-integrable function $ \mathbb{R} \rightarrow \mathbb{R}$. The total integral of $ f(x)$ has to be $ 1$.

To create a discrete representation of a PDF, we use the fact that the probability of an interval is given by its integral. Basically, there are two ways to approximate a PDF, which mainly differ in the way how the intervals are determined. The first one uses a sampling rate and, therefore, a fixed interval size. The second one uses arbitrary sizes for intervals. Both methods store the probabilities for the intervals, not the probability density.



Subsections

Snowball 2007-03-16