2 The test

We consider the cases for which distributions of sources are assumed to be known. In these cases, a probability $p\{\xi <x\}$ is well defined on every event $\{\xi <x\}$. This is the probability that the event is found in the range $\xi <x$, where $\xi$ is the random variable. The cumulative distribution function of the random variable $\xi$ for a given distribution is defined as

$$f(x)\equiv p\{\xi <x\}.$$ (1)

For a sample with size N, the frequency of events with $\{\xi <x\}$, $p^{*}\{\xi <x\}$, is used to define the distribution function

$$f_N(x)\equiv p^{*}\{\xi <x\}.$$ (2)

Let the sample be $S\equiv \{x_i\vert i\in I\}$, $I\equiv \{i\vert 1\leq i\leq N\}$, where $x_i\leq x_{i+1}$, $\forall i,i+1\in I$. According to the above definition, we have

$$f_N(x)=\left\{ \begin{array} {c} 0 \\ \frac iN \\ 1 \en... ...1) \\ (x_i<x&\leq x_{i+1}) \\ (x_N&<x) \end{array}.\right.$$ (3)

According to (1), giving a cumulative distribution function f(x), the probability of event $\{\xi <x\}$ is known. This is f(x) itself. For a sample, the number of events with $\{\xi <x\}$, N_x, follows the well-known binomial distribution, with

$$E\{N_x\}=Nf(x)$$ (4)

and

$${\rm Var}\{N_x\}=Nf(x)[1-f(x)],$$ (5)

where N is the size of the sample. According to (2), the distribution function of the sample is

$$f_N(x)=\frac{N_x}N.$$ (6)

It is clear that f_N(x) is simply the percentage of the x_i's values less than x in the sample of size N.

Substituting (6) into (4) and (5) we have

$$E\{f_N(x)\}=\frac{E\{N_x\}}N=f(x)$$ (7)

and

$${\rm Var}\{f_N(x)\}=\frac{{\rm Var}\{N_x\}}{N^2}={\frac{f(x)[1-f(x)]}N}.$$ (8)

Since the variance of the distribution function is known for any given value of the random variable, we are able to compare observational data with the given distribution at any data point. This leads to the following definition.

Definition 1. For a sample with size N, if the following condition

$$\left\vert f_N(x)-f(x)\right\vert <{\sqrt{{\frac{f(x)[1-f(x)]}N}}}$$ (9)

is satisfied for any given x, the sample is said to pass the $1\sigma$distribution function deviation test for the distribution of f(x).

It is clear that this method concerns the individual, rather than the collective, behavior of samples. By comparing the distribution function of a sample with the given cumulative distribution function one can tell if the sample obeys the given distribution at the $1\sigma$confidence level using Eq. (9).