Maz values of Unit Bounded Distributions or Mixing Distributions

Source: vignettes/Mixing_Distributions_mazxxx.Rmd

IT WOULD BE CLEARLY BENEFICIAL FOR YOU BY USING THE RMD FILES IN THE GITHUB DIRECTORY FOR FURTHER EXPLANATION OR UNDERSTANDING OF THE R CODE FOR THE RESULTS OBTAINED IN THE VIGNETTES.

Moment about zero values for all six Unit Bounded Distributions

Using Moment about zero values is useful to calculating mean, variance, skewness and kurtosis. There is no useful need to plot the moment about zero values against how shape parameter values change. Therefore, I have not plotted it.

Below are the six functions which can produce moment about zero values.

  • mazUNI - producing moment about zero values for Uniform distribution.
  • mazTRI - producing moment about zero values for Triangular distribution.
  • mazBETA - producing moment about zero values for Beta distribution.
  • mazKUM - producing moment about zero values for Kumaraswamy distribution.
  • mazGHGBeta - producing moment about zero values for Gaussian Hyper-geometric Generalized Beta distribution.
  • mazGBeta1 - producing moment about zero values for Generalized Beta Type 1 distribution.
  • mazGamma - producing moment about zero values for Gamma distribution.

Consider the \(r^{th}\) Moment about zero for the Beta distribution for when a random variable \(P\) is given below

\[E[P^r]= \prod_{i=0}^{r-1} \frac{(\alpha+i)}{(\alpha+\beta+i)} \] where \(\alpha\) (a) and \(\beta\) (b) are shape parameters(\(\alpha, \beta > 0\)) .

  • The mean is acquired by \[\mu= E[P]\]
  • The variance is acquired by \[ \sigma_2= E[P^2] - \mu^2 \]
  • The skewness is acquired by \[\gamma_1= \frac{E[(P- \mu)^3]}{(Var(P))^{(3/2)}} \]
  • The kurtosis is acquired by \[ \gamma_2= \frac{E[(P- \mu)^4]}{(Var(P))^{2}}\]

Using the above four equations it is possible to find the mean, variance, skewness and kurtosis. It is even possible to validate the mean and variance calculated from dxxx functions through the mazxxx functions.

Proving Mean is similar from dxxx and mazxxx functions

Mean from dBETA compared with mazBETA function

## Mean from dBETA function for (a=3, b=9) = 0.25
## Mean from mazBETA function for (a=3, b=9) = 0.25

Proving Variance is similar from dxxx and mazxxx functions

Variance from dBETA compared with mazBETA function

## Variance from dBETA function for (a=3,b=9) = 0.01442308
## Variance from mazBETA function for (a=3,b=9) = 0.01442308

Conclusion

According to the above outputs it clear that the mean and variance can be acquired using moment about zero value functions.