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Let \(X_1, X_2, \ldots, X_n\) be independent and identically distributed \({\it Normal}(0, 1)\) variables. Then $$ X^2 = \sum_{i=1}^n X_i^2 $$
is said to follow a chi-square distribution with n degrees of freedom.
Returns the value at x of the density function of a Chi-square random variable \(\chi^2(n)\), with n>0. The \(\chi^2(n)\) random variable is equivalent to the \(\Gamma\left(n/2,2\right)\).
The pdf is
$$ f(x; n) = \cases{ \displaystyle{x^{n/2-1} e^{-x/2} \over 2^{n/2} \Gamma\left(\displaystyle{n\over 2}\right)} & for $x > 0$ \cr \cr 0 & otherwise } $$(%i1) load ("distrib")$
(%i2) pdf_chi2(x,n);
n/2 - 1 - x/2
x %e unit_step(x)
(%o2) -----------------------------
n n/2
gamma(-) 2
2
Returns the value at x of the distribution function of a Chi-square random variable \(\chi^2(n)\), with n>0.
The cdf is $$ F(x; n) = \cases{ 1 - Q\left(\displaystyle{n\over 2}, {x\over 2}\right) & $x > 0$ \cr 0 & otherwise } $$
where Q(a,z) is the gamma_incomplete_regularized function.
(%i1) load ("distrib")$
(%i2) cdf_chi2(3,4);
3
(%o2) 1 - gamma_incomplete_regularized(2, -)
2
(%i3) float(%); (%o3) 0.4421745996289252
Returns the q-quantile of a Chi-square random variable
\(\chi^2(n)\), with n>0; in other words, this is the inverse of cdf_chi2. Argument q must be an element of [0,1].
This function has no closed form and it is numerically computed.
(%i1) load ("distrib")$
(%i2) quantile_chi2(0.99,9); (%o2) 21.66599433346194
Returns the mean of a Chi-square random variable \(\chi^2(n)\), with n>0.
The \(\chi^2(n)\) random variable is equivalent to the \(\Gamma\left(n/2,2\right)\).
The mean is $$ E[X] = n $$
(%i1) load ("distrib")$
(%i2) mean_chi2(n); (%o2) n
Returns the variance of a Chi-square random variable \(\chi^2(n)\), with n>0.
The \(\chi^2(n)\) random variable is equivalent to the \(\Gamma\left(n/2,2\right)\).
The variance is $$ V[X] = 2n $$
(%i1) load ("distrib")$
(%i2) var_chi2(n); (%o2) 2 n
Returns the standard deviation of a Chi-square random variable \(\chi^2(n)\), with n>0.
The \(\chi^2(n)\) random variable is equivalent to the \(\Gamma\left(n/2,2\right)\).
The standard deviation is $$ D[X] = \sqrt{2n} $$
(%i1) load ("distrib")$
(%i2) std_chi2(n); (%o2) sqrt(2) sqrt(n)
Returns the skewness coefficient of a Chi-square random variable \(\chi^2(n)\), with n>0.
The \(\chi^2(n)\) random variable is equivalent to the \(\Gamma\left(n/2,2\right)\).
The skewness coefficient is $$ SK[X] = \sqrt{8\over n} $$
(%i1) load ("distrib")$
(%i2) skewness_chi2(n);
3/2
2
(%o2) -------
sqrt(n)
Returns the kurtosis coefficient of a Chi-square random variable \(\chi^2(n)\), with n>0.
The \(\chi^2(n)\) random variable is equivalent to the \(\Gamma\left(n/2,2\right)\).
The kurtosis coefficient is $$ KU[X] = {12\over n} $$
(%i1) load ("distrib")$
(%i2) kurtosis_chi2(n);
12
(%o2) --
n
Returns a Chi-square random variate
\(\chi^2(n)\), with n>0. Calling random_chi2 with a second argument m, a random sample of size m will be simulated.
The simulation is based on the Ahrens-Cheng algorithm. See random_gamma for details.
To make use of this function, write first load("distrib").