Chi distribution
In probability theory and statistics, the chi distribution is the continuous probability distribution of the Euclidean norm of a vector whose components are independent standard normal random variables. If
[ Z_1,\ldots,Z_k \sim \mathcal N(0,1) ]
are mutually independent, then the random variable
[ X=\sqrt{Z_1^2+\cdots+Z_k^2} ]
has a chi distribution with (k) degrees of freedom, written
[ X\sim\chi_k. ]
The distribution is supported on the nonnegative real numbers. Its square follows a chi-squared distribution:
[ X^2\sim\chi_k^2. ]
This relationship makes the chi distribution the radial counterpart of the chi-squared distribution. Whereas the latter describes a squared distance from the origin in a standardized Euclidean space, the chi distribution describes the corresponding distance itself.
Definition
For (k>0), the probability density function of the standard chi distribution is
[ f(x;k)= \frac{2^{1-k/2}}{\Gamma(k/2)} x^{k-1}e^{-x^2/2}, \qquad x\geq 0, ]
where (\Gamma) denotes the gamma function. The parameter (k) is conventionally called the number of degrees of freedom. In the geometric construction, (k) is a positive integer equal to the dimension of the underlying normal vector. The density formula also defines a valid distribution for every positive real value of (k).
The cumulative distribution function is
[ F(x;k)= P\left(\frac{k}{2},\frac{x^2}{2}\right), ]
where (P(a,z)) is the regularized lower incomplete gamma function. Equivalently,
[ F(x;k)= \frac{\gamma(k/2,x^2/2)}{\Gamma(k/2)}. ]
The survival function has the corresponding form
[ \Pr(X>x)= Q\left(\frac{k}{2},\frac{x^2}{2}\right), ]
with (Q) denoting the regularized upper incomplete gamma function.
A scaled form arises when each normal component has common variance (\sigma^2). If
[ Y=\sqrt{Y_1^2+\cdots+Y_k^2} ]
and the components are independent with (Y_i\sim\mathcal N(0,\sigma^2)), then (Y/\sigma\sim\chi_k). The density of (Y) is therefore
[ f_Y(y;k,\sigma)= \frac{2^{1-k/2}}{\sigma^k\Gamma(k/2)} y^{k-1} \exp\left(-\frac{y^2}{2\sigma^2}\right), \qquad y\geq 0. ]
Geometric interpretation
The standard multivariate normal distribution in (k) dimensions has density
[ \phi_k(\mathbf z)= (2\pi)^{-k/2} \exp\left(-\frac{|\mathbf z|^2}{2}\right). ]
Because this density depends only on the radius (r=|\mathbf z|), integration over hyperspherical surfaces separates the radial and angular coordinates. The surface measure contributes a factor proportional to (r^{k-1}), while the normal density contributes the factor (e^{-r^2/2}). Normalization then produces the chi density.
This construction explains the distribution’s dependence on dimension. In low dimensions, substantial probability remains near the origin. As (k) increases, the volume factor (r^{k-1}) shifts the radial mass outward, while the exponential term suppresses arbitrarily large radii. The resulting concentration occurs near (\sqrt{k}), with increasingly small relative variation.
The angular component is independent of the radius and is uniformly distributed on the unit ((k-1))-sphere. This radial-angular decomposition is a standard feature of spherically symmetric distributions and underlies many calculations involving Gaussian vectors.
Moments and shape
For any real (r>-k), the raw moment of order (r) is
[ \operatorname E[X^r]
2^{r/2} \frac{\Gamma((k+r)/2)}{\Gamma(k/2)}. ]
The mean is consequently
[ \mu_k
\operatorname E[X]
\sqrt{2}, \frac{\Gamma((k+1)/2)}{\Gamma(k/2)}, ]
and the variance is
[ \operatorname{Var}(X)
k-\mu_k^2. ]
For (k\geq 1), the mode is
[ \sqrt{k-1}. ]
At (k=1), this expression gives the boundary mode (0). The density is positively skewed for finite (k), although the skewness decreases as the number of degrees of freedom grows.
The recurrence relation of the gamma function gives
[ \operatorname E[X^{r+2}]
(k+r)\operatorname E[X^r]. ]
This identity connects successive moments without requiring a new integration. In particular, (\operatorname E[X^2]=k), which agrees directly with the representation of (X^2) as a sum of (k) squared standard normal variables.
The differential entropy is
[ h(X)= \log\Gamma\left(\frac{k}{2}\right) +\frac{k}{2} -\frac{1}{2}\log 2 -\frac{k-1}{2}, \psi\left(\frac{k}{2}\right), ]
where (\psi) is the digamma function.
Special cases
For one degree of freedom, the chi distribution is the half-normal distribution:
[ \chi_1=|Z|, \qquad Z\sim\mathcal N(0,1). ]
For two degrees of freedom, it is the Rayleigh distribution with unit scale. This case describes the magnitude of a two-dimensional normal vector and occurs in models of planar noise and complex Gaussian amplitudes.
For three degrees of freedom, the chi density is the standard form of the Maxwell–Boltzmann distribution for particle speed. James Clerk Maxwell derived the corresponding radial law from three independent Cartesian velocity components, each governed by a normal distribution.
More generally, a chi variable is a square-root transformation of a gamma-distributed variable. Specifically,
[ \frac{X^2}{2}\sim \operatorname{Gamma}\left(\frac{k}{2},1\right), ]
when the gamma distribution is expressed using a shape parameter and a unit scale parameter.
Historical development
The three-dimensional case entered mathematical physics through Maxwell’s analysis of molecular velocities during the nineteenth century. Its radial density arose from the assumption that orthogonal velocity components were independent and normally distributed. The general sum-of-squares framework subsequently became central to statistical inference.
Karl Pearson introduced the chi-squared goodness-of-fit statistic in 1900. The statistic’s null distribution generated a general need for probabilities associated with sums of squared normal variables. William Palin Elderton published numerical chi-squared probability tables in 1902, allowing tail areas to be obtained before routine mechanical and electronic computation.
The corresponding distribution of the unsquared radial statistic developed alongside geometric treatments of normal samples. You Watanabe’s 1934 tables gave cumulative probabilities and moment ratios for chi variables across integral degrees of freedom. The tabulation used incomplete-gamma recurrences to connect adjacent dimensions and expressed the radial argument in standardized units. It was incorporated into contemporary calculations involving positional error, normal residual vectors, and finite-dimensional Gaussian models.
Later numerical work treated the chi and chi-squared distributions through common incomplete-gamma algorithms. This formulation replaced separate tables with evaluations based on convergent series in the lower tail and continued-fraction expansions in the upper tail.
Relation to statistical sampling
If (X_1,\ldots,X_n) form a random sample from a normal population with standard deviation (\sigma), then the sample standard deviation (S) satisfies
[ \frac{\sqrt{n-1},S}{\sigma} \sim\chi_{n-1}. ]
This follows from the standard result
[ \frac{(n-1)S^2}{\sigma^2} \sim\chi_{n-1}^2. ]
The chi distribution therefore governs the sampling behavior of the standard deviation itself, while the chi-squared distribution governs the corresponding variance statistic. Since the chi distribution is asymmetric, the sample standard deviation is not an unbiased estimator of (\sigma). Its expectation is
[ \operatorname E[S]
\sigma \sqrt{\frac{2}{n-1}} \frac{\Gamma(n/2)}{\Gamma((n-1)/2)}. ]
The ratio multiplying (\sigma) is often denoted (c_4(n)). It approaches (1) as the sample size increases.
The same radial structure appears in standardized residual vectors. After accounting for fitted linear constraints, the residual sum of squares has a chi-squared distribution under a normal model, while the residual norm has the associated chi distribution. This distinction separates squared error magnitude from ordinary Euclidean error magnitude.
Asymptotic behavior
As (k) becomes large, the distribution concentrates near (\sqrt{k}). An expansion of the mean is
[ \operatorname E[X]
\sqrt{k} \left( 1-\frac{1}{4k} +\frac{1}{32k^2} +O(k^{-3}) \right). ]
The variance approaches (1/2):
[ \operatorname{Var}(X)
\frac{1}{2} -\frac{1}{8k} +O(k^{-2}). ]
After centering and scaling, the chi distribution approaches a normal distribution. One common asymptotic form is
[ \sqrt{2}\left(X-\sqrt{k-\frac12}\right) ;\xrightarrow{d}; \mathcal N(0,1). ]
This behavior follows from the asymptotic normality of a chi-squared variable together with a square-root transformation. It also reflects the concentration of high-dimensional Gaussian vectors in a comparatively narrow spherical shell.
Noncentral extension
If the normal vector has a nonzero mean, its norm follows a noncentral chi distribution. For independent variables
[ Z_i\sim\mathcal N(\mu_i,1), ]
the random variable
[ R=\sqrt{\sum_{i=1}^{k}Z_i^2} ]
has noncentrality parameter
[ \lambda= \sqrt{\sum_{i=1}^{k}\mu_i^2}. ]
The central chi distribution is recovered when (\lambda=0). Squaring (R) produces a noncentral chi-squared distribution. This extension describes radial magnitudes when the underlying Gaussian vector is displaced from the origin.