Statistics/Chi-squared distribution

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The probability density function (pdf) of the chi-squared distribution as used in statistics is

$f(x;\,k)={\dfrac {x^{{\frac {k}{2}}-1}e^{-{\frac {x}{2}}}}{2^{\frac {k}{2}}\Gamma \left({\frac {k}{2}}\right)}}$, where ${\textstyle x>0}$

Here, ${\textstyle \Gamma (k/2)}$ denotes the Gamma_function.

The use of the gamma function in the equation below reflects the chi-squared distribution's origin as a special case of the gamma distribution.

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distribution.

The probability density function (pdf) of the gamma distribution is given by the formula

$f(x;k,\theta )={\frac {x^{k-1}e^{-x/\theta }}{\theta ^{k}\Gamma (k)}}\quad {\text{ for }}x>0{\text{ and }}k,\theta >0.$

where Γ(k) is the Gamma_function, with shape parameter k and a scale parameter θ.

The cumulative probability distribution of the gamma distribution is the area under the curve of the ditribution, which indicates the increasing probability of the x value of a single random point within the gamma distribution being less than or equal to the x value of the cumulative probability distribution. The gamma cumulative probability distribution function can be calculated as

$F(x;k,\theta )=\int _{0}^{x}f(u;k,\theta )\,du={\frac {\gamma \left(k,{\frac {x}{\theta }}\right)}{\Gamma (k)}},$

where $\gamma \left(k,{\frac {x}{\theta }}\right)$ is the lower incomplete gamma function.

The lower incomplete gamma function can be calculated as

$$x^{s}\,\Gamma (s)\,e^{-x}\sum _{m=0}^{\infty }{\frac {x^{m}}{\Gamma (s+m+1)}}$$

and so, for the chi-squared cumulative probability distribution with chi-squared k, we have, substituting chi-square k into s as k/2,

$$F(x;\,k)=x^{(k/2)}\,\Gamma (k/2)\,e^{-x}\sum _{m=0}^{\infty }{\frac {x^{m}}{\Gamma ({\frac {k}{2}}+m+1)}}.$$