# Chi-square distribution

Parameters Probability density function Cumulative distribution function  $k > 0\,$ degrees of freedom $x \in [0; +\infty)\,$ $\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,$ $\frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,$ $k\,$ approximately $k-2/3\,$ $k-2\,$ if $k\geq 2\,$ $2\,k\,$ $\sqrt{8/k}\,$ $12/k\,$ $\frac{k}{2}\!+\!\ln(2\Gamma(k/2))\!+\!(1\!-\!k/2)\psi(k/2)$ $(1-2\,t)^{-k/2}$ for $2\,t<1\,$ $(1-2\,i\,t)^{-k/2}\,$

In probability theory and statistics, the chi-square distribution (also chi-squared or χ2  distribution) is one of the most widely used theoretical probability distributions in inferential statistics, e.g., in statistical significance tests. It is useful because, under reasonable assumptions, easily calculated quantities can be proven to have distributions that approximate to the chi-square distribution if the null hypothesis is true.

The best-known situations in which the chi-square distribution are used are the common chi-square tests for goodness of fit of an observed distribution to a theoretical one, and of the independence of two criteria of classification of qualitative data. Many other statistical tests also lead to a use of this distribution, like Friedman's analysis of variance by ranks.

## Definition

If Xi are k independent, normally distributed random variables with mean 0 and variance 1, then the random variable $Q = \sum_{i=1}^k X_i^2$

is distributed according to the chi-square distribution with k degrees of freedom. This is usually written $Q\sim\chi^2_k.\,$

The chi-square distribution has one parameter: k - a positive integer that specifies the number of degrees of freedom (i.e. the number of Xi)

The chi-square distribution is a special case of the gamma distribution.

## Characteristics

### Probability density function

A probability density function of the chi-square distribution is $f(x;k)= \begin{cases}\displaystyle \frac{1}{2^{k/2}\Gamma(k/2)}\,x^{(k/2) - 1} e^{-x/2}&\text{for }x>0,\\ 0&\text{for }x\le0, \end{cases}$

where Γ denotes the Gamma function, which has closed-form values at the half-integers.

### Cumulative distribution function $F(x;k)=\frac{\gamma(k/2,x/2)}{\Gamma(k/2)} = P(k/2, x/2)$

where γ(k,z) is the lower incomplete Gamma function and P(k,z) is the regularized Gamma function.

Tables of this distribution — usually in its cumulative form — are widely available and the function is included in many spreadsheets and all statistical packages.

### Characteristic function

The characteristic function of the Chi-square distribution is $\chi(t;k)=(1-2it)^{-k/2}.\,$

### Expected value and variance

If $X\sim\chi^2_k$ then the mean is given by $\frac{}{} \mathrm{E}(X)=k,$

and the variance is given by $\frac{}{} \mathrm{Var}(X)=2k.$

### Median

The median of $X\sim\chi^2_k$ is given approximately by $k-\frac{2}{3}+\frac{4}{27k}-\frac{8}{729k^2}.$

### Information entropy

The information entropy is given by $H = \int_{-\infty}^\infty f(x;k)\ln(f(x;k)) dx = \frac{k}{2} + \ln \left( 2 \Gamma \left( \frac{k}{2} \right) \right) + \left(1 - \frac{k}{2}\right) \psi(k/2).$

where ψ(x) is the Digamma function.

### Noncentral moments

The moments about zero of a chi-square distribution with k degrees of freedom are given by \begin{align} E(X^m) &= k (k+2) (k+4) \cdots (k+2m-2) \\ &= 2^m \frac{\Gamma(m+k/2)}{\Gamma(k/2)}. \end{align}

### Derivation of the pdf for one degree of freedom

Let random variable Y be defined as Y = X2 where X has normal distribution with mean 0 and variance 1 (that is X ~ N(0,1)).

Then if $y<0, ~ P(Y and if $y\geq0, ~ P(Y $f_y(y) = f_x(\sqrt{y})\frac{\partial(\sqrt{y})}{\partial y}-f_x(-\sqrt{y})\frac{\partial(-\sqrt{y})}{\partial y}$ $= \frac{1}{\sqrt{2\pi}}e^{\frac{-y}{2}}\frac{1}{2y^{1/2}} + \frac{1}{\sqrt{2\pi}}e^{\frac{-y}{2}}\frac{1}{2y^{1/2}}$ $= \frac{1}{2^{\frac{1}{2}} \Gamma(\frac{1}{2})}y^{\frac{1}{2} -1}e^{\frac{-y}{2}}$

Then $Y = X^2 \sim \chi^2_1$.

## Related distributions and properties

The chi-square distribution has numerous applications in inferential statistics, for instance in chi-square tests and in estimating variances. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a regression line via its role in Student's t-distribution. It enters all analysis of variance problems via its role in the F-distribution, which is the distribution of the ratio of two independent chi-squared random variables divided by their respective degrees of freedom.

• If $X\sim\chi^2_k$, then as k tends to infinity, the distribution of $(X-k)/\sqrt{2k}$ tends to a standard normal distribution: see asymptotic distribution. This follows directly from the definition of the chi-squared distribution, the central limit theorem, and the fact that the mean and variance of $\chi^2_1$ are 1 and 2 respectively. However, convergence is slow as the skewness is $\sqrt{8/k}$ and the excess kurtosis is 12 / k.
• If $X\sim\chi^2_k$ then $\sqrt{2X}$ is approximately normally distributed with mean $\sqrt{2k-1}$ and unit variance (result credited to R. A. Fisher).
• If $X\sim\chi^2_k$ then $\sqrt{X/k}$ is approximately normally distributed with mean 1 − 2 / (9k) and variance 2 / (9k) (Wilson and Hilferty,1931)
• $X \sim \mathrm{Exponential}(\lambda = \tfrac{1}{2})$ is an exponential distribution if $X \sim \chi_2^2$ (with 2 degrees of freedom).
• $Y \sim \chi_{\nu}^2$ is a chi-square distribution if $Y = \sum_{m=1}^{\nu} X_m^2$ for $X_i \sim N(0,1)$ independent that are normally distributed.
• If $\boldsymbol{z}'=[Z_1,Z_2,\cdots,Z_n]$, where the Zis are independent Normal(0,σ2) random variables or $\boldsymbol{z}\sim N_p(\boldsymbol{0},\sigma^2 \mathrm{I})$ and $\boldsymbol{A}$ is an $n\times n$ idempotent matrix with rank nk then the quadratic form $\frac{\boldsymbol{z}'\boldsymbol{A}\boldsymbol{z}}{\sigma^2}\sim \chi^2_{n-k}$.
• If the $X_i\sim N(\mu_i,1)$ have nonzero means, then $Y = \sum_{m=1}^k X_m^2$ is drawn from a noncentral chi-square distribution.
• The chi-square distribution $X\sim\chi^2_\nu$ is a special case of the gamma distribution, in that $X \sim {\Gamma}(\frac{\nu}{2}, \theta=2)$.
• $Y \sim \mathrm{F}(\nu_1, \nu_2)$ is an F-distribution if $Y = \frac{X_1 / \nu_1}{X_2 / \nu_2}$ where $X_1 \sim \chi_{\nu_1}^2$ and $X_2 \sim \chi_{\nu_2}^2$ are independent with their respective degrees of freedom.
• $Y \sim \chi^2(\bar{\nu})$ is a chi-square distribution if $Y = \sum_{m=1}^N X_m$ where $X_m \sim \chi^2(\nu_m)$ are independent and $\bar{\nu} = \sum_{m=1}^N \nu_m$.
• if X is chi-square distributed, then $\sqrt{X}$ is chi distributed.
• in particular, if $X \sim \chi_2^2$ (chi-square with 2 degrees of freedom), then $\sqrt{X}$ is Rayleigh distributed.
• if $X_1, \dots, X_n$ are i.i.d. N(μ,σ2) random variables, then $\sum_{i=1}^n(X_i - \bar X)^2 \sim \sigma^2 \chi^2_{n-1}$ where $\bar X = \frac{1}{n} \sum_{i=1}^n X_i$.
• if $X \sim \mathrm{SkewLogistic}(\tfrac{1}{2})\,$, then $\mathrm{log}(1 + e^{-X}) \sim \chi_2^2\,$
• The box below shows probability distributions with name starting with chi for some statistics based on $X_i\sim \mathrm{Normal}(\mu_i,\sigma^2_i),i=1,\cdots,k,$ independent random variables:
Name Statistic
chi-square distribution $\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2$
noncentral chi-square distribution $\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2$
chi distribution $\sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}$
noncentral chi distribution $\sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}$