Maz values of Unit Bounded Distributions or Mixing Distributions

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 themoment 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 zerovalues.

• mazUNI - producing moment about zero values for Uniformdistribution.
• mazTRI - producing moment about zero values forTriangular distribution.
• mazBETA - producing moment about zero values for Betadistribution.
• mazKUM - producing moment about zero values forKumaraswamy distribution.
• mazGHGBeta - producing moment about zero values forGaussian Hyper-geometric Generalized Beta distribution.
• mazGBeta1 - producing moment about zero values forGeneralized Beta Type 1 distribution.
• mazGamma - producing moment about zero values for Gammadistribution.

Consider the\(r^{th}\) Moment aboutzero 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 themean and variance calculated from dxxx functions through the mazxxxfunctions.

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 variancecan be acquired using moment about zero value functions.