Chi Distribution

The chi (χ) distribution models the distribution of the square root of the sum of the squares of n independent normal variables. It is related to the chi-squared distribution, which models the sum of the squares of n independent normal variables. The chi distribution is used in various statistical applications, including hypothesis testing and confidence interval estimation.

The chi distribution with n degrees of freedom represents the distribution of the norm of a vector of n independent normal variables. The probability density function (PDF) is given by:

The cumulative distribution function (CDF) is:

where $\mathrm{\Gamma }$ is the Gamma function and $\gamma$ is the lower incomplete Gamma function. The distribution is defined for non-negative real values (x ≥ 0), and while the degrees of freedom parameter n is typically a positive integer, it can be any positive real number.

The chi distribution plays a fundamental role in various fields of statistics and physics, with numerous practical applications.

• In statistical quality control, the chi distribution serves as a crucial tool for monitoring process variability and constructing control charts that help maintain manufacturing quality standards.

• Signal processing engineers utilize the chi distribution for analyzing the envelope detection of Gaussian noise signals, which is essential in communication systems and radar applications.

• The field of medical imaging relies on the chi distribution for analyzing magnetic resonance imaging (MRI) data, particularly in characterizing noise patterns and improving image quality.

• In physics, the Maxwell-Boltzmann distribution, which is a special case of the chi distribution with three degrees of freedom, provides a mathematical model for describing the distribution of molecular speeds in ideal gases at thermal equilibrium.

The chi distribution has several important properties, including:

The chi distribution is closely related to the chi-squared distribution. The square root of a variable with a chi-squared distribution has a chi distribution. Special cases of the chi distribution include the Rayleigh distribution (2 degrees of freedom) and the Maxwell-Boltzmann distribution (3 degrees of freedom).

The chi distribution is implemented by the ChiDistribution class. It has one constructor which takes the degrees of freedom as its only argument. The following constructs a chi distribution with 10 degrees of freedom:

var chi = new ChiDistribution(10);

Dim chi = New ChiDistribution(10)

No code example is currently available or this language may not be supported.

let chi = ChiDistribution(10.0)

The ChiDistribution class has one specific property, DegreesOfFreedom, that returns the degrees of freedom of the distribution.

ChiDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator.

var random = new Pcg32();
double sample = ChiDistribution.Sample(random, 10);

Dim random = New Pcg32()
Dim sample = ChiDistribution.Sample(random, 10)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = ChiDistribution.Sample(random, 10.0)

The above example uses the Pcg32 to generate uniform random numbers.

For more information on the chi distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 1", Chapter 18. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 11. Wiley Series in Probability and Statistics.

Other Resources