Page 105 - Standard Handbook Of Petroleum & Natural Gas Engineering

94 Mathematics
Moment Ratios

The moment ratios are dimensionless coefficients used to compare characteristics
of distributions measured on different scales.
The coefficient of variation is a measure of relative dispersion of a set of values
and is given for the population by

and for the sample by

C" = s/x

The coefficient of skewness is a measure of relative symmetry of a distribution
and is given for the population by

and for the sample by

The coefficient of kurtosis is a measure of relative peakedness and is given by
P, = k/S4

Common Probability Distributions for Continuous Random Variables
The parameters of a distribution control its geometric characteristics [23]:

1. A location parameter is the abscissa of a location point and may be a measure
of central tendency, such as a mean.
2. A scale parameter determines the location of fractiles of the distribution
relative to some specified point, often the value of the location parameter.
3. Shape parameters control the geometric configuration of a distribution.
There may be zero, one, or multiple shape parameters.

A bounded continuous random variable with uniform distribution has the
probability function
f(X) = I $'(P-a) a < X < P
otherwise

where a = location parameter, representing lower limit of the distribution
P = scale parameter, representing upper bound of the distribution

Probabilities are determined by integration over the necessary range, Le.,
P(X, < x < X,) = p/(p - a)dX
I