|
| "Distribution Functions" |
|
| Chi and Chi-Square Distribution Method |
|
|
Chi Distribution Results |
|
|
Chi-Square Distribution Results |
|
IMPLEMENTATION
Chi and Chi-Square Distribution Method
The Chi and Chi-Square Distributions are continous probability distributions. The distribution usually arises when an
n-dimensional vector's orthogonal components are independent and each follows a standard
normal distribution.
Probability Density Function
The Chi (or ε) probability density function is expressed in terms of the formula:
|
f(x;n,δ) = [ 2 (n/2)n/2 xn-1 en/2δx2 /
γ(n/2) εn
|
where γ is the gamma function.
The Chi square (or ε2) probability density function is defined by
|
f(x;n,δ) = xn/2 - 1 e n/2γ2 * x2 /
2n/2 γ(n/2) γn
|
Chi and Chi-Square Random Number Generators
The Chi square random distribution with parameters n and γ is equal to the sum
squares of n independent normal random distribution:
|
γ2 (n,γ) = ∑i=1n [Ni (0,γ2)]2
|
The Chi random distribution can be calculated from the above Chi square random distribution:
Testing the Chi and Chi-Square Distribution Method
To test the Chi and Chi-Square Distribution method, a new static method has been added. The TestChiAndChi-SquareDistribution() method has
been written and executed. No additional code is shown. The user can change variables as desired.
For the test, two parameters were set:
number of bins = 20; (nBins)
number of points = 2000; (nPoints)
|
where the parameter (nBins) is the number of bins in the histogram and (nPoints)
is the number of random points. A random array is created using the Chi and Chi-Square distribution. A comparison is made between
the histogram of random data and the theoretical probability density function of the Chi and Chi-Square distribution. One can see the
results from the exponential random generator are very close to the theoretical Chi and Chi-Square distribution function.
Running this example generates the results shown above.
static void TestChiAndChi-SquareDistribution();
{
for (int i = 0; i < nBins; i++)
{
ListBox1.Items.Add(" x = " + xdata[i] + "," + " - - - - -> Random Data = " + ydata1[i] + "," + " - - - - -> Density Distribution = " + Math.Round(ychi[i] * normalizeFactor1, 0).ToString());
}
}
{
for (int i = 0; i < nBins; i++)
{
ListBox2.Items.Add(" x = " + xdata[i] + "," + " - - - - -> Random Data = " + ydata2[i] + "," + " - - - - -> Density Distribution = " + Math.Round(ychisquare[i] * normalizeFactor2, 0).ToString());
}
}
|
|
|
|