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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
$$ \sum_{k=1}^n \mu_k^2 $$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 $$ f(x; n, \lambda) = {1\over 2}e^{-(x+\lambda)/2} \left(x\over \lambda\right)^{n/4-1/2}I_{{n\over 2} - 1}\left(\sqrt{n \lambda}\right) $$
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").
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.
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.
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.
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.
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.
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.
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").
Next: F Random Variable, Previous: Chi-squared Random Variable, Up: Functions and Variables for continuous distributions [Contents][Index]