is often shown with a delta function representation. A novice reader relying on integral properties of the delta function may found this notation quite confusing. Probably, the notation itself is an example of abuse of notation.
One dimensional function/distribution: Sum of delta functions
Let's define a one dimensional function, $f(x)$ as follows, $x \in \mathbb{R}$ and $a$ being constant:
$ f(x) = a \sum_{i=-n}^{n} \delta(x - x_{i})$
This representation is inspired from Dirac comb and used in spike trains. Note that set of data points in one dimension $\{x_{i} \}$ will determine the graph of this function. Using the shifting property of delta function, the value of the function will be zero every where except on data points. The constant $a$ will simply be the height of the graph at the data point.
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| Figure: A spike train. |
Numeric Example
Let's plot $f(x)$ for some specific values of the set $\{x_{i} \} = {-0.5, -0.2, -0.1, 0.2, 0.4}$ and $a=0.5$. Here is the R code for plotting this spike train.
x_i = c(-0.5, -0.2, -0.1, 0.2, 0.4) a = c(0.5, 0.5, 0.5, 0.5, 0.5) plot(x_i,a,type="h",xlim=c(-0.6,0.6),ylim=c(0,0.6),lwd=2,col="blue",ylab="p")
Representing Histograms: One dimensional example
Particularly convenient representation of histograms can be developed similarly. Consider set of points $\{x_{i}\}_{i=1}^{n}$ where we would like to establish a histogram out of this set, let's say $h(x)$. If we set our histogram intervals as $\{x_{j}\}_{j=1}^{m}$. The histogram $h(x)$ then can be written as
$h(x_{j}) = \sum_{i=1}^{n} \sum_{j=1}^{m} \delta(x_{j}- x_{i}^{min})$
where set $x_{i}^{min}$ represents the value from set $\{x_{j}\}_{j=1}^{m}$ that is closest to given $x_{i}$. Where as, second sum determines the height at a given point, i.e., frequency. This is just a confusing mathematical representation and practical implementation only counts the frequency of $x_{i}^{min}$ directly.
Conclusion
However it is quite trivial, the above usage of sum of delta functions appear in mathematical physics as well, not limited to statistics.
