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Chi distribution

The chi distribution, denoted \chi(k), is a continuous probability distribution in probability theory and statistics, representing the distribution of the Euclidean norm (or positive square root) of a k-dimensional vector whose components are independent standard normal random variables, where k > 0 is the degrees of freedom parameter.^[1]^[2] Its probability density function is given by

f(x; k) = \frac{2^{1 - k/2} x^{k-1} e^{-x^2/2}}{\Gamma(k/2)}, \quad x > 0,

where \Gamma denotes the gamma function, and the distribution is supported on the positive real line.^[2]^[3] The mean of the chi distribution is \mu = \sqrt{2} \cdot \frac{\Gamma((k+1)/2)}{\Gamma(k/2)}, while the variance is \sigma^2 = k - \mu^2.^[2]^[3] The mode occurs at \sqrt{k-1} for k > 1, and higher moments can be expressed using the gamma function as E[X^r] = 2^{r/2} \cdot \frac{\Gamma((k+r)/2)}{\Gamma(k/2)}.^[2] The cumulative distribution function lacks a simple closed form but can be expressed using the regularized lower incomplete gamma function: F(x; k) = P\left(\frac{k}{2}, \frac{x^2}{2}\right), where P(a, z) = \frac{\gamma(a, z)}{\Gamma(a)} and \gamma is the lower incomplete gamma function.^[4] Special cases of the chi distribution include the half-normal distribution when k=1, the Rayleigh distribution (with scale parameter 1) when k=2, and the Maxwell–Boltzmann distribution when k=3.^[1] It is closely related to the chi-squared distribution, as the square of a chi-distributed random variable follows a chi-squared distribution with k degrees of freedom.^[4] In applications, the chi distribution appears in multivariate statistics for modeling the magnitude of vectors from normal populations, in signal processing via the Rayleigh case for envelope detection of narrowband signals, and in physics through the Maxwell–Boltzmann form for speeds of ideal gas particles; it also supports reliability analysis and numerical methods for finding critical values in hypothesis testing.^[1]^[4]

Introduction

Definition and Interpretation

The chi distribution with k degrees of freedom is the probability distribution of the positive square root of a chi-squared random variable with k degrees of freedom.^[5] This arises naturally in statistical contexts where the magnitude of deviations or errors is of interest, transforming the non-negative chi-squared values into a distribution supported on the positive reals.^[6] A key interpretation of the chi distribution is as the Euclidean norm (length) of a k-dimensional vector whose components are independent standard normal random variables.^[6] Geometrically, it represents the distance from the origin to a point in k-dimensional Euclidean space, where the point is randomly sampled from a multivariate normal distribution centered at the origin with identity covariance matrix. This connection provides intuition for applications in fields like signal processing and physics, where such norms model radial distances in high-dimensional noise.^[6]

Parameters and Support

The chi distribution is parameterized by a single positive real number k > 0, which represents the degrees of freedom and can take non-integer values.^[1]^[7] A random variable X following the chi distribution with parameter k, denoted X \sim \chi(k), has support on the interval [0, \infty), with probability zero for all x < 0.^[7]^[8] As a continuous probability distribution, P(X = 0) = 0 holds for any k > 0.^[1] The probability density approaches zero as x \to \infty.^[8] The standard form of the chi distribution assumes unit scale, arising as the Euclidean norm of a k-dimensional vector of independent standard normal random variables; more general variants include non-central and scaled chi distributions, though these are not parameterized in the standard case.^[1]^[7]

Mathematical Formulation

Probability Density Function

The probability density function (PDF) of the chi distribution with k degrees of freedom is given by

f(x; k) = \frac{2^{1 - k/2} x^{k-1} e^{-x^2 / 2}}{\Gamma(k/2)}, \quad x \geq 0,

and f(x; k) = 0 otherwise, where \Gamma denotes the gamma function and k > 0^[8]^[9]. This PDF arises as the distribution of the Euclidean norm of a k-dimensional vector of independent standard normal random variables, or equivalently, as the square root of a chi-squared random variable with k degrees of freedom. To derive it, consider a random variable Y \sim \chi^2(k) with PDF

g(y; k) = \frac{y^{k/2 - 1} e^{-y/2}}{2^{k/2} \Gamma(k/2)}, \quad y \geq 0.

Apply the transformation X = \sqrt{Y}, so Y = X^2 and the Jacobian is |dy/dx| = 2x. The PDF of X is then

f(x; k) = g(x^2; k) \cdot 2x = \frac{(x^2)^{k/2 - 1} e^{-x^2/2}}{2^{k/2} \Gamma(k/2)} \cdot 2x = \frac{2^{1 - k/2} x^{k-1} e^{-x^2 / 2}}{\Gamma(k/2)},

for x \geq 0^[9]^[10]. The formula is normalized such that \int_0^\infty f(x; k) \, dx = 1. To verify, substitute z = x^2, so dz = 2x \, dx and x \, dx = dz/2, yielding

\int_0^\infty f(x; k) \, dx = \int_0^\infty \frac{z^{k/2 - 1} e^{-z/2}}{2^{k/2} \Gamma(k/2)} \, dz = \frac{1}{\Gamma(k/2)} \int_0^\infty \frac{z^{k/2 - 1} e^{-z/2}}{2^{k/2}} \, dz.

The integral is the gamma function definition scaled: \int_0^\infty z^{k/2 - 1} e^{-z/2} \, dz = 2^{k/2} \Gamma(k/2), confirming the total probability is 1^[8]^[9]. The shape of the PDF varies with k: for small k (e.g., $1 < k < 2), it is right-skewed and increasing to a mode before decreasing; as k increases, the distribution becomes less skewed and more symmetric, approaching a normal distribution for large k (e.g., k \geq 90)^[10].

Cumulative Distribution Function

The cumulative distribution function (CDF) of the chi distribution with k degrees of freedom is given by

F(x; k) = \frac{\gamma\left(\frac{k}{2}, \frac{x^2}{2}\right)}{\Gamma\left(\frac{k}{2}\right)},

for x \geq 0, where \gamma(s, z) denotes the lower incomplete gamma function and \Gamma(s) is the gamma function.^[4]^[8] This expression arises because the chi distribution is the distribution of the positive square root of a chi-squared random variable with k degrees of freedom, so F(x; k) equals the CDF of the chi-squared distribution evaluated at x^2.^[4] An equivalent form uses the upper incomplete gamma function \Gamma(s, z):

F(x; k) = 1 - \frac{\Gamma\left(\frac{k}{2}, \frac{x^2}{2}\right)}{\Gamma\left(\frac{k}{2}\right)},

since \gamma(s, z) + \Gamma(s, z) = \Gamma(s).^[8] The gamma form is commonly used due to the direct connection to the chi-squared distribution. To derive the CDF, integrate the probability density function of the chi distribution from 0 to x:

F(x; k) = \int_0^x f(t; k) \, dt,

where f(t; k) is the PDF. Substitute u = t^2 / 2, so du = t \, dt and t^{k-1} \, dt = 2^{(k/2)-1} u^{(k/2)-1} \, du. This transforms the integral into the form of the CDF of a gamma distribution with shape parameter k/2 and scale 2 (equivalent to the chi-squared CDF at x^2), yielding the incomplete gamma expression above.^[4]^[8] The quantile function, or inverse CDF F^{-1}(p; k), has no closed-form expression and must be computed numerically, often by inverting the chi-squared quantile and taking the square root or using root-finding methods on the incomplete gamma function.^[4] This numerical inversion is commonly employed in statistical software for generating random variates from the chi distribution via the inverse transform sampling method.^[8]

Generating Functions

The moment-generating function (MGF) of the chi distribution with k degrees of freedom is

M(t; k) = \, _1F_1\left(\frac{k}{2}; \frac{1}{2}; \frac{t^2}{2}\right) + t\sqrt{2} \frac{\Gamma\left(\frac{k+1}{2}\right)}{\Gamma\left(\frac{k}{2}\right)} \, _1F_1\left(\frac{k+1}{2}; \frac{3}{2}; \frac{t^2}{2}\right),

where _1F_1(\cdot; \cdot; \cdot) denotes Kummer's confluent hypergeometric function of the first kind. This expression is obtained by evaluating the defining integral M(t; k) = \mathbb{E}[e^{tX}] = \int_0^\infty e^{tx} f(x; k) \, dx, where f(x; k) is the probability density function of the chi distribution; substituting the PDF and performing the integration yields the hypergeometric form through known integral representations of the confluent hypergeometric function. The MGF is defined for t in the interval where the integral converges, consistent with the distribution's support on [0, \infty). The characteristic function (CF) of the chi distribution is similarly given by

\phi(t; k) = \, _1F_1\left(\frac{k}{2}; \frac{1}{2}; -\frac{t^2}{2}\right) + it\sqrt{2} \frac{\Gamma\left(\frac{k+1}{2}\right)}{\Gamma\left(\frac{k}{2}\right)} \, _1F_1\left(\frac{k+1}{2}; \frac{3}{2}; -\frac{t^2}{2}\right),

where i = \sqrt{-1} is the imaginary unit; this follows analogously from the integral definition \phi(t; k) = \mathbb{E}[e^{itX}] = \int_0^\infty e^{itx} f(x; k) \, dx, leading to the same hypergeometric structure via series expansion.

Properties

Moments and Cumulants

The raw moments of the chi distribution with k > 0 degrees of freedom are given by

\mu_n = E[X^n] = 2^{n/2} \frac{\Gamma\left(\frac{k + n}{2}\right)}{\Gamma\left(\frac{k}{2}\right)},

where \Gamma denotes the gamma function.^[10] This expression arises from the integral representation of the expectation using the probability density function and properties of the gamma function.^[10] The mean, or first raw moment, is

\mu = E[X] = \sqrt{2} \frac{\Gamma\left(\frac{k + 1}{2}\right)}{\Gamma\left(\frac{k}{2}\right)}.

^[8] The second raw moment is E[X^2] = k, since X^2 follows a chi-squared distribution with k degrees of freedom.^[10] Consequently, the variance is

\sigma^2 = \operatorname{Var}(X) = k - \mu^2.

^[10] These central moments characterize the location and scale of the distribution, with the mean increasing with k and the variance reflecting the spread relative to the chi-squared parent. The third raw moment is E[X^3] = 2\sqrt{2} \frac{\Gamma\left(\frac{k + 3}{2}\right)}{\Gamma\left(\frac{k}{2}\right)}.^[8] The skewness \gamma_1 = \frac{E[(X - \mu)^3]}{\sigma^3} is positive for all k > 0, indicating right-skewness, and decreases toward zero as k increases, reflecting the distribution's approach to symmetry. The third central moment E[(X - \mu)^3] is computed using the raw moments via E[X^3] - 3\mu k + 2\mu^3.^[8] The fourth raw moment is E[X^4] = k(k + 2).^[8] The excess kurtosis \gamma_2 = \frac{E[(X - \mu)^4]}{\sigma^4} - 3 is positive for finite k and approaches 0 as k \to \infty, consistent with the central limit theorem's normal approximation for large degrees of freedom. The fourth central moment E[(X - \mu)^4] is computed using the raw moments via E[X^4] - 4\mu E[X^3] + 6\mu^2 k - 3\mu^4.^[8] The cumulants \kappa_r of the chi distribution are derived from the cumulant-generating function, \log M(t), where M(t) is the moment-generating function.^[10] Only the first two cumulants have simple closed forms: \kappa_1 = \mu (the mean) and \kappa_2 = \sigma^2 (the variance); higher-order cumulants lack elementary expressions and require evaluation via the full moment-generating function, which involves modified Bessel functions.^[8]

Entropy

The differential entropy H(X) of a continuous random variable X with probability density function f(x) is given by the integral

H(X) = -\int_0^\infty f(x) \ln f(x) \, dx.

For the chi distribution with k > 0 degrees of freedom, the PDF is

f(x) = \frac{2^{1 - k/2} x^{k-1} e^{-x^2/2}}{\Gamma(k/2)}, \quad x > 0.

Substituting this into the entropy integral yields

\ln f(x) = (1 - k/2) \ln 2 + (k-1) \ln x - x^2/2 - \ln \Gamma(k/2).

The entropy then becomes

H(X) = -(1 - k/2) \ln 2 - (k-1) \mathbb{E}[\ln X] + \frac{1}{2} \mathbb{E}[X^2] + \ln \Gamma(k/2).

Using the known moments and properties of the distribution, with \mathbb{E}[X^2] = k and \mathbb{E}[\ln X] = \frac{1}{2} \left( \psi\left( \frac{k}{2} \right) + \ln 2 \right), where \psi is the digamma function, this simplifies to

H(X) = \frac{k}{2} - \frac{1}{2} \ln 2 + \ln \Gamma\left( \frac{k}{2} \right) - \frac{k-1}{2} \psi\left( \frac{k}{2} \right).

These expressions are derived by evaluating the expectations through integration by parts and leveraging the integral representations of the gamma and digamma functions, such as \mathbb{E}[\ln X] expressed via the digamma function after substitution and change of variables relating to the gamma distribution. The entropy H(X) increases with k, as larger degrees of freedom broaden the support and increase the uncertainty of the distribution. For the special case k=2 (corresponding to the Rayleigh distribution), H(X) \approx 0.942.

Mode, Median, and Approximations

The mode of the chi distribution with parameter k > 1 is located at \sqrt{k - 1}, obtained by setting the derivative of the probability density function to zero, which yields the equation x^2 = k - 1. For $0 < k \leq 1, the mode is at 0, as the density is monotonically decreasing. This location closely approximates the mean for large k.^[11]^[12] The median of the chi distribution has no closed-form expression and must generally be computed numerically from the cumulative distribution function. For practical purposes, an approximation for the median m is m \approx \sqrt{k - 2/3}, which provides reasonable accuracy even for moderate k. This follows from applying the square root to the corresponding approximation for the median of the related chi-squared distribution.^[13] For large values of k, the chi distribution is well-approximated by a normal distribution via the central limit theorem applied to the sum of squares of the underlying standard normal variables, or equivalently through the delta method transformation of the chi-squared approximation. Specifically, a chi-distributed random variable X satisfies X \approx \mathcal{N}\left(\sqrt{k - 1/2}, 1/2\right), where the mean \sqrt{k - 1/2} refines the leading-order term \sqrt{k} and the variance approaches $1/2 independently of k. This normal approximation is particularly useful for tail probabilities and confidence intervals when k \gtrsim 30. The median and mode also align closely with this mean under the approximation.^[14]^[13] The upper tail behavior for fixed k as x \to \infty follows from asymptotic analysis of the incomplete gamma function in the cumulative distribution, yielding

P(X > x) \sim \frac{x^{k-2} e^{-x^2/2}}{2^{k/2-1} \Gamma(k/2)}.

This Mills ratio-like approximation arises because the hazard rate of the distribution is asymptotically x, making the survival function comparable to the density divided by x. It is effective for computing large-deviation probabilities without numerical integration.^[15]

Connection to Chi-Squared

The chi distribution with k degrees of freedom arises as the distribution of the square root of a random variable following a central chi-squared distribution with the same k degrees of freedom. Specifically, if Y \sim \chi^2(k), then X = \sqrt{Y} \sim \chi(k).^[10]^[16] This relationship can be established through a change-of-variable transformation on the probability density function. Assuming the density of Y is known, the density of X is obtained by substituting y = x^2 and applying the Jacobian factor of $2x, yielding the standard form of the chi density after simplification.^[10] The parameter k, representing degrees of freedom in both distributions, corresponds to the dimensionality underlying their constructions from normal variables. For the chi-squared distribution, Y is the sum of squares of k independent standard normal random variables, Y = \sum_{i=1}^k Z_i^2 where Z_i \sim N(0,1). Consequently, for the chi distribution, X = \sqrt{Y} = \left( \sum_{i=1}^k Z_i^2 \right)^{1/2} represents the Euclidean norm (or root-mean-square magnitude) of the same k-dimensional standard normal vector.^[10]^[16] This connection extends to the noncentral case, where if Y \sim \chi^2(k, \lambda) follows a noncentral chi-squared distribution with k degrees of freedom and noncentrality parameter \lambda > 0, then X = \sqrt{Y} follows a noncentral chi distribution with parameters k and \lambda. However, the central case (\lambda = 0) is the primary focus here, reducing to the standard chi and chi-squared distributions. In statistical inference, the chi-squared distribution underpins chi-squared tests for goodness-of-fit, independence, and variance, leveraging its asymptotic properties under the null hypothesis. In contrast, the chi distribution emerges in contexts involving root-mean-square errors or magnitudes of normal errors, such as in signal processing or measurement precision analysis, where the square root transformation normalizes squared error aggregates.^[16]^[16]

Special Cases and Generalizations

The chi distribution with one degree of freedom is known as the half-normal distribution, which arises as the absolute value of a standard normal random variable, and its probability density function is given by f(x) = \sqrt{\frac{2}{\pi}} e^{-x^2/2}, \quad x \geq 0. ^[17] For two degrees of freedom, the chi distribution corresponds to the [Rayleigh distribution](/page/Rayleigh distribution), commonly applied in signal processing for modeling the magnitude of vectors with normally distributed components, with probability density function f(x) = x e^{-x^2/2}, \quad x \geq 0. ^[18] The case of three degrees of freedom yields the Maxwell–Boltzmann distribution (up to a scale factor), which describes the speeds of particles in an ideal gas, and its probability density function is f(x) = \sqrt{\frac{2}{\pi}} x^2 e^{-x^2/2}, \quad x \geq 0. ^[19] A key generalization is the non-central chi distribution, which extends the standard form to the Euclidean norm of a multivariate normal vector with non-zero mean, accounting for a non-centrality parameter that shifts the distribution.^[20] The scaled chi distribution incorporates a scale parameter σ > 0 by multiplying the standard chi random variable by σ, adjusting the distribution for varying variances in the underlying normals.^[19] For integer degrees of freedom k, the cumulative distribution function of the chi distribution admits closed-form simplifications: even k values involve finite sums of Poisson probabilities, while odd k values incorporate the error function alongside exponential terms.^[19]

Applications and Estimation

Applications in Statistics and Physics

In statistics, the chi distribution arises as the probability distribution of the sample standard deviation when sampling from a normal population, where the scaled sample standard deviation follows a chi distribution with n-1 degrees of freedom, specifically \sqrt{(n-1)} s / \sigma \sim \chi_{n-1}, with s denoting the sample standard deviation and \sigma the population standard deviation.^[21] This connection enables the construction of confidence intervals for the population variance using the related chi-squared distribution, as the square of the scaled sample standard deviation yields a chi-squared random variable, allowing bounds on \sigma^2 via the square root transformation for standard deviation intervals.^[22] In physics, the chi distribution with three degrees of freedom describes the Maxwell–Boltzmann speed distribution for the speeds of particles in an ideal gas at thermal equilibrium, where the probability density function for molecular speeds aligns with the chi form scaled by temperature and mass parameters.^[23] In wireless communications, the Rayleigh distribution—a special case of the chi distribution with two degrees of freedom—models the envelope of received signals in Rayleigh fading channels, capturing amplitude variations due to multipath propagation without a dominant line-of-sight path.^[24] In engineering applications, the chi distribution characterizes root-mean-square (RMS) noise levels in systems with additive white Gaussian noise, as the RMS value of a Gaussian process follows a scaled chi distribution, aiding in noise floor predictions and system performance analysis.^[25] For radar systems, detection thresholds are set using the chi distribution to model signal envelopes in Gaussian clutter, particularly for fluctuating targets where the probability of detection is computed from chi-distributed amplitudes to maintain constant false alarm rates.^[26] Historically, the chi distribution relates indirectly to Karl Pearson's chi-squared test, introduced in 1900 for assessing goodness-of-fit in biometric data, where the square root of the test statistic provides a measure of effect size magnitude in early 20th-century statistical biometrics.^[27]

Parameter Estimation Methods

The maximum likelihood estimator (MLE) for the degrees of freedom parameter k of the chi distribution is obtained by maximizing the log-likelihood function \ell(k) = \sum_{i=1}^n \log f(x_i; k), where f(x; k) is the probability density function of the chi distribution. This leads to the transcendental equation involving the digamma function:

\psi\left(\frac{k}{2}\right) = \frac{1}{n} \sum_{i=1}^n \log(x_i^2) - \ln 2.

Since no closed-form solution exists, numerical methods such as Newton-Raphson iteration are required to solve for \hat{k}.^[28] The method of moments provides an alternative approach by equating the theoretical mean of the chi distribution to the sample mean. The mean is given by \mathbb{E}[X] = \sqrt{2} \frac{\Gamma((k+1)/2)}{\Gamma(k/2)}, so the estimating equation is

\bar{x} = \sqrt{2} \frac{\Gamma((k+1)/2)}{\Gamma(k/2)},

which must be solved iteratively for \hat{k}, often using initial approximations like \hat{k} \approx \bar{x}^2. For refinement, a second moment can be matched, leveraging the fact that \mathbb{E}[X^2] = k, yielding \hat{k} = \frac{1}{n} \sum_{i=1}^n x_i^2, which is unbiased but less efficient than MLE for small samples.^[28] The MLE \hat{k} is biased downward for small sample sizes n, with the bias becoming negligible as n increases. An approximate bias correction can be applied using higher-order expansions involving the trigamma function. The asymptotic variance of \hat{k} is \frac{4}{n \trigamma(k/2)}, derived from the inverse Fisher information, providing standard errors for inference. For large k, this approximates \frac{2k}{n}.^[28] To assess the fit of the estimated chi distribution, goodness-of-fit tests such as the Kolmogorov-Smirnov (KS) test can be employed. The KS test compares the empirical cumulative distribution function (CDF) of the sample to the theoretical CDF F(x; \hat{k}) of the chi distribution, with the test statistic D_n = \sup_x | \hat{F}_n(x) - F(x; \hat{k}) |, which under the null hypothesis converges to a known distribution for large n. Rejection occurs if D_n exceeds critical values from Kolmogorov's distribution.^[28]

References

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### Summary of scipy.stats.chi
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