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

$$f(x;n,\lambda )=\frac{1}{2}{e}^{-(x+\lambda )/2}{\left(\frac{x}{\lambda }\right)}^{n/4-1/2}{I}_{\frac{n}{2}-1}\left(\sqrt{n\lambda }\right)$$

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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").

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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

$$SK[X]=\frac{2^{3/2}(n+3\mu )}{(n+2\mu {)}^{3/2}}$$

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

$$KU[X]=\frac{12(n+4\mu )}{(2+2\mu {)}^2}$$

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").

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