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Chi and Chi-Square Distribution Method
Chi Distribution Results

Chi-Square Distribution Results

Number of Points:
Number of Bins:

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

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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