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 "Distribution Functions" Chi and Chi-Square Distribution Method
 Chi Distribution Results x = 0.1, - - - - -> Random Data = 87, - - - - -> Density Distribution = 75 x = 0.3, - - - - -> Random Data = 215, - - - - -> Density Distribution = 208 x = 0.5, - - - - -> Random Data = 317, - - - - -> Density Distribution = 295 x = 0.7, - - - - -> Random Data = 318, - - - - -> Density Distribution = 325 x = 0.9, - - - - -> Random Data = 325, - - - - -> Density Distribution = 303 x = 1.1, - - - - -> Random Data = 257, - - - - -> Density Distribution = 249 x = 1.3, - - - - -> Random Data = 194, - - - - -> Density Distribution = 182 x = 1.5, - - - - -> Random Data = 133, - - - - -> Density Distribution = 120 x = 1.7, - - - - -> Random Data = 81, - - - - -> Density Distribution = 72 x = 1.9, - - - - -> Random Data = 45, - - - - -> Density Distribution = 39 x = 2.1, - - - - -> Random Data = 12, - - - - -> Density Distribution = 19 x = 2.3, - - - - -> Random Data = 8, - - - - -> Density Distribution = 9 x = 2.5, - - - - -> Random Data = 8, - - - - -> Density Distribution = 4 x = 2.7, - - - - -> Random Data = 0, - - - - -> Density Distribution = 1 x = 2.9, - - - - -> Random Data = 0, - - - - -> Density Distribution = 0 x = 3.1, - - - - -> Random Data = 0, - - - - -> Density Distribution = 0 x = 3.3, - - - - -> Random Data = 0, - - - - -> Density Distribution = 0 x = 3.5, - - - - -> Random Data = 0, - - - - -> Density Distribution = 0 x = 3.7, - - - - -> Random Data = 0, - - - - -> Density Distribution = 0 x = 3.9, - - - - -> Random Data = 0, - - - - -> Density Distribution = 0

 Chi-Square Distribution Results x = 0.1, - - - - -> Random Data = 192, - - - - -> Density Distribution = 192 x = 0.3, - - - - -> Random Data = 167, - - - - -> Density Distribution = 174 x = 0.5, - - - - -> Random Data = 168, - - - - -> Density Distribution = 157 x = 0.7, - - - - -> Random Data = 130, - - - - -> Density Distribution = 142 x = 0.9, - - - - -> Random Data = 127, - - - - -> Density Distribution = 129 x = 1.1, - - - - -> Random Data = 132, - - - - -> Density Distribution = 116 x = 1.3, - - - - -> Random Data = 97, - - - - -> Density Distribution = 105 x = 1.5, - - - - -> Random Data = 112, - - - - -> Density Distribution = 95 x = 1.7, - - - - -> Random Data = 76, - - - - -> Density Distribution = 86 x = 1.9, - - - - -> Random Data = 68, - - - - -> Density Distribution = 78 x = 2.1, - - - - -> Random Data = 70, - - - - -> Density Distribution = 71 x = 2.3, - - - - -> Random Data = 59, - - - - -> Density Distribution = 64 x = 2.5, - - - - -> Random Data = 55, - - - - -> Density Distribution = 58 x = 2.7, - - - - -> Random Data = 54, - - - - -> Density Distribution = 52 x = 2.9, - - - - -> Random Data = 52, - - - - -> Density Distribution = 47 x = 3.1, - - - - -> Random Data = 32, - - - - -> Density Distribution = 43 x = 3.3, - - - - -> Random Data = 40, - - - - -> Density Distribution = 39 x = 3.5, - - - - -> Random Data = 41, - - - - -> Density Distribution = 35 x = 3.7, - - - - -> Random Data = 35, - - - - -> Density Distribution = 32 x = 3.9, - - - - -> Random Data = 21, - - - - -> Density Distribution = 29

 Number of Points: Number of Bins: [ Initial Number of Bins: {20) ] [ Initial Number of Points: {2000} ]

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:

 γ(n,ε) = √ γ2 (n,ε)/ n   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());
}
}

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expertise..."

EIGENVALUE
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