52.2.4 Chi-squared Random Variable

Let \(X_1, X_2, \ldots, X_n\) be independent and identically distributed \({\it Normal}(0, 1)\) variables. Then

is said to follow a chi-square distribution with \(n\) degrees of freedom.

Function: pdf_chi2 (x,n) ¶

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

Categories: Package distrib ·

Function: cdf_chi2 (x,n) ¶

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

Categories: Package distrib ·

Function: quantile_chi2 (q,n) ¶

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

Categories: Package distrib ·

Function: mean_chi2 (n) ¶

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

Categories: Package distrib ·

Function: var_chi2 (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

Categories: Package distrib ·

Function: std_chi2 (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)

Categories: Package distrib ·

Function: skewness_chi2 (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)

Categories: Package distrib ·

Function: kurtosis_chi2 (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

Categories: Package distrib ·

Function: random_chi2 (n)
    random_chi2 (n,m) ¶

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").

Categories: Package distrib · Random numbers ·