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

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

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.

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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.

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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.

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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.

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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.

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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.

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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 ·