#### 52.2.5 Noncentral Chi-squared Random Variable

Let \(X_1, X_2, ..., X_n\) be n independent normally distributed random variables with means \(\mu_k\) and unit variances. Then the random variable

$$ \sum_{k=1}^n X_k^2 $$

has a noncentral \(\chi^2\) distribution. The number of degrees of freedom is n, and the noncentrality parameter is defined by

$$ \sum_{k=1}^n \mu_k^2 $$

Function: pdf_noncentral_chi2 (x,n,ncp)

  Returns the value at x of the density function of a noncentral \(\chi^2\) random variable m4_noncentral_chi2(n,ncp), with n>0 and noncentrality parameter ncp>=0. To make use of this function, write first `load("distrib")`.

  For x < 0, the pdf is 0, and for \(x \ge 0\) the pdf is $$ f(x; n, \lambda) = {1\over 2}e^{-(x+\lambda)/2} \left(x\over \lambda\right)^{n/4-1/2}I_{{n\over 2} - 1}\left(\sqrt{n \lambda}\right) $$

  Categories:Package distrib ·

Function: cdf_noncentral_chi2 (x,n,ncp)

  Returns the value at x of the distribution function of a noncentral Chi-square random variable m4_noncentral_chi2(n,ncp), with n>0 and noncentrality parameter ncp>=0. To make use of this function, write first `load("distrib")`.

  Categories:Package distrib ·

Function: quantile_noncentral_chi2 (q,n,ncp)

  Returns the q-quantile of a noncentral Chi-square random variable m4_noncentral_chi2(n,ncp), with n>0 and noncentrality parameter ncp>=0; in other words, this is the inverse of `cdf_noncentral_chi2`. Argument q must be an element of [0,1].

  This function has no closed form and it is numerically computed.

  Categories:Package distrib ·

Function: mean_noncentral_chi2 (n,ncp)

  Returns the mean of a noncentral Chi-square random variable m4_noncentral_chi2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

  The mean is $$ E[X] = n + \mu $$

  where \(\mu\) is the noncentrality parameter ncp.

  Categories:Package distrib ·

Function: var_noncentral_chi2 (n,ncp)

  Returns the variance of a noncentral Chi-square random variable m4_noncentral_chi2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

  The variance is $$ V[X] = 2(n+2\mu) $$

  where \(\mu\) is the noncentrality parameter ncp.

  Categories:Package distrib ·

Function: std_noncentral_chi2 (n,ncp)

  Returns the standard deviation of a noncentral Chi-square random variable m4_noncentral_chi2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

  The standard deviation is $$ D[X] = \sqrt{2(n+2\mu)} $$

  where \(\mu\) is the noncentrality parameter ncp.

  Categories:Package distrib ·

Function: skewness_noncentral_chi2 (n,ncp)

  Returns the skewness coefficient of a noncentral Chi-square random variable m4_noncentral_chi2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

  The skewness coefficient is $$ SK[X] = {2^{3/2}(n+3\mu) \over (n+2\mu)^{3/2}} $$

  where \(\mu\) is the noncentrality parameter ncp.

  Categories:Package distrib ·

Function: kurtosis_noncentral_chi2 (n,ncp)

  Returns the kurtosis coefficient of a noncentral Chi-square random variable m4_noncentral_chi2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

  The kurtosis coefficient is $$ KU[X] = {12(n+4\mu)\over (2+2\mu)^2} $$

  where \(\mu\) is the noncentrality parameter ncp.

  Categories:Package distrib ·

Function: random_noncentral_chi2 (n,ncp)
`random_noncentral_chi2` (n,ncp,m)

  Returns a noncentral Chi-square random variate m4_noncentral_chi2(n,ncp), with n>0 and noncentrality parameter ncp>=0. Calling `random_noncentral_chi2` with a third argument m, a random sample of size m will be simulated.

  To make use of this function, write first `load("distrib")`.

  Categories:Package distrib ·Random numbers ·