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Sampling - Fixed intervals

To create an approximation of a PDF by a set of fixed intervals, the domain of the PDF is devided into $ N$ intervals denoted by the set $ I$, each of which has the same width specified by the value $ d$. The $ i$th interval is then defined by $ [(i-1/2)d, (i + 1/2)d[$. For our purposes, we can assume that the domain of a PDF is always greater or equal to zero. Thus, we set the first interval ($ i=0$) to $ [0, 1/2 d[$. To minimize computational errors, we associate the probability of the $ i$th interval $ [(i-1/2)d, (i + 1/2)d[$ to its middle value $ i * d$. So, we get a set of $ N$ probabilities, where the probability of interval $ i$, $ p_i$ is given by the integral:

$\displaystyle p_i = \int_{(i-1/2)d}^b f(x) dx, \qquad lim_{b \rightarrow (i + 1/2)d}
$

for $ i>0$ and

$\displaystyle p_i = \int_{0}^b f(x) dx, \qquad lim_{b \rightarrow 1/2 d}
$

for $ i=0$. The approximation of a PDF is completely described by the interval width $ d$ and the probabilities $ p_i$ for all intervals $ I$.

Example 2.1  
Figure 2.12: The probability of the interval $ [2.5d, 3.5d[$ is the striped area under the graph.
Image SamplePDF

Figure 2.12 illustrates how a pdf $ f(x)$ is approximated by deviding its domain into a set of intervals. The X-axis shows the multiples of the interval width $ d$ ($ 1d$, $ 2d$, $ 3d$...). These values represent the mean values of the intervals to which the probabilities will be associated. The figure shows how the probability of the third interval is computed. The interval borders are given by $ [2.5d, 3.5d[$ and its mean value is $ 3d$. The probability $ p_3$ is the striped area under the graph. So, all values lying in this area are associated to the value $ 3d$.


next up previous contents index
Next: Approximation by boxes - Up: PDF discretisation Previous: PDF discretisation   Contents   Index
Snowball 2007-03-16