NON CENTRAL CHI SQUARE DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.NonCentralChiSquare({"lambda": *, "n": *})

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

## CDF, PDF, PPF receive float or numpy.ndarray.
distribution.cdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.pdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.ppf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.sample(int) # -> numpy.ndarray

## STATS
distribution.mean # -> float
distribution.variance # -> float
distribution.standard_deviation # -> float
distribution.skewness # -> float
distribution.kurtosis # -> float
distribution.median # -> float
distribution.mode # -> float

Equations

Distribution Definition

$$X\sim \mathrm{NonCentralChiSquare}(\lambda ,n)$$

Distribution Domain

$$x\in [0,+\mathrm{\infty })$$

Parameters Domain and Constraints

$$\lambda \in {\mathbb{R}}^{+},n\in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=1-Q_{\frac{n}{2}}(\sqrt{\lambda },\sqrt{x})$$

Probability Density Function

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

Percent Point Function / Sample

$${\text{Sample}}_X=\sum _{i=1}^n{(\sqrt{\frac{\lambda }{n}}+{\mathrm{\Phi }}^{-1}\left(u_i\right))}^2$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)dx=2^{k-1}(k-1)!(n+k\lambda )+\sum _{j=1}^{k-1}\frac{(k-1)!2^{j-1}}{(k-j)!}(n+j\lambda ){\mu }_{k-j}^{\prime }$$

Parametric Mean

$$\mathrm{Mean}(X)={\mu }_1^{\prime }=n+\lambda$$

Parametric Variance

$$\mathrm{Variance}(X)={\mu }_2^{\prime }-{\mu }_1^{\prime 2}=2(n+2\lambda )$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=\frac{2^{3/2}(n+3\lambda )}{(n+2\lambda {)}^{3/2}}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}=\frac{12(n+4\lambda )}{(n+2\lambda {)}^2}$$

Parametric Median

$$\mathrm{Median}(X)=F_X^{-1}\left(\frac{1}{2}\right)$$

Parametric Mode

$$\mathrm{Mode}(X)=\arg \underset{x}{max}{f}_X\left(x\right)$$

Additional Information and Definitions

• $$\begin{gathered} \text{Computing an analytic expression for the inverse of the cumulative distribution function is not} \\ \text{feasible. Nonetheless, it is possible to generate a random sample from the distribution.} \end{gathered}$$
• $u_i:\text{Uniform[0,1] random varible}$
• ${\mathrm{\Phi }}^{-1}\left(x\right):\text{PPF normal standard distribution}$
• $I_{\alpha }\left(x\right):\text{Modified Bessel function of the first kind of order }\alpha \in \mathbb{N}$

Spreadsheet Documents

• Phitter playground
• Download Excel Spreadsheet
• Excel file from GitHub repository
• Google spreadsheet document