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Evaluate the Pearson IV probability density function (PDF) for real-valued argument x

$$\frac{1}{f(x)}\frac{df}{dx}=\frac{(x-\lambda)+a_{0}}{b_{0}+b_{1}(x-\lambda)+b_{2}(x-\lambda)^{2}}$$

The Pearson PDF is categorized as type IV when the discriminant b0+b1x+b2x2 has no real roots. The Pearson IV PDF is typically defined in terms of the coefficients a0,b0 ,b1 and b2 that depend on the variance σ2, skewness γ1, and kurtosis β2. Additionally, the location parameter λ is related to the mean

$$\lambda=\mu-\frac{\gamma_{1}\sigma}{4}(r-2)$$

and the mode (maximum)

$$M=\mu-\frac{\gamma_{1}\sigma}{2}(\frac{r-2}{r+2})$$

Parameter r is defined as

$$r=2m-2=\frac{6(\beta_2-\gamma_1^2-1)}{2\beta_2-3\gamma_1^2-6}$$

The Pearson IV distribution requires:

$$0\leq \gamma_1^2<32$$

and

$$\beta_2>\frac{39\gamma_1^2+6(\gamma_1^2+4)^{3/2}+48}{32-\gamma_1^2}$$

such that the PDF has a finite mean, variance, skewness and kurtosis. When γ1=0 and β2=3, the Pearson IV PDF reduces to the normal distribution.

 Note: γ1=0 and β2=3 is not supported by the last condition and will result in an error when calling the pearson4pdf command

 Supported Product: FDTD, MODE, DEVICE, INTERCONNECT

 Syntax Description f = pearson4pdf(x) Returns thePearson IV probability density function (PDF) for real-valued argument x. f = pearson4pdf(x,mu,sigma,gamma1,beta2) Returns thePearson IV probability density function (PDF) for real-valued argument x. Please see above for the definition of µ, σ, γ1, and β2(β2=3+δ).