Distributional Property of a Proportion Trait Whose Component Traits Follow a Normal Distribution

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Abstract

This paper concerns the characteristics of the distribution of a proportion trait that is defined as the proportion of the amount of one component to the total amount of all the components. Two component traits following curtailed normality are considered here and are assumed to have positive values. The probability density function for the proportion trait is derived in the general form and then the particular form is given for the case where coefficients of variation for the component traits are relatively small. Using three selected sets of the parameter values for the component traits, the parametric values of the location, dispersion, skewness and kurtosis for the distribution of the proportion trait are shown. It is revealed that the shape of the distribution is able to be symmetrical and very nearly normal, skewed to the right or skewed to the left, with a density more peaked around its center than a normal curve. These results confirm us that phenotypic selection differentials in one cycle of upward and downward truncation selection for the proportion trait may be asymmetrical, especially with higher intensity of selection, even when the component traits follow a normal distribution.

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