Statistics/Chi-squared distribution

Statistics/Chi-squared distribution is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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 distribution, 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,

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

In practice, this series formula is often subject to accumulated errors from rounding in the frequently used region where x and k are under 10 and near one another. You may therefore instead use a mathematics function library, if available for your programming task, to calculate gamma and incomplete gamma.

Task

• Calculate and show the values of the χ2(x; k) for k = 1 through 5 inclusive and x integer from 0 and through 10 inclusive.

• Create a function to calculate the cumulative probability function for the χ2 distribution. This will need to be reasonably accurate (at least 6 digit accuracy) for k = 3.

• Calculate and show the p values of statistical samples which result in a χ2(k = 3) value of 1, 2, 4, 8, 16, and 32. (Statistical p values can be calculated for the purpose of this task as approximately 1 - P(x), with P(x) the cumulative probability function at x for χ2.)

The following is a chart for 4 airports:

Flight Delays

Airport	On Time	Delayed	Totals
Dallas/Fort Worth	77	23	100
Honolulu	88	12	100
LaGuardia	79	21	100
Orlando	81	19	100
All Totals	325	75	400
Expected per 100	81.25	18.75	100

χ2 on a 2D table is calculated as the sum of the squares of the differences from expected divided by the expected numbers for each entry on the table. The k for the chi-squared distribution is to be calculated as df, the degrees of freedom for the table, which is a 2D parameter, (count of airports - 1) * (count of measures per airport - 1), which here is (4 - 1 )(2 - 1) = 3.

• Calculate the Chi-squared statistic for the above table and find its p value using your function for the cumulative probability for χ2 with k = 3 from the previous task.

Stretch task

• Show how you could make a plot of the curves for the probability distribution function χ2(x; k) for k = 0, 1, 2, and 3.