Page 117 - Standard Handbook Of Petroleum & Natural Gas Engineering

102 Mathematics

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If the probabilities do not remain constant over the trials and if there are k
(rather than two) possible outcomes of each trial, the hypergeometric distribution
applies. For a sample of size N of a population of size T, where

t, + t, + . . . + t, = T, and
n, + n2 + . . . + nt = N

the probability is

The Poisson distribution can be used to determine probabilities for discrete
random variables where the random variable is the number of times that an
event occurs in a single trial (unit of time, space, etc.). The probability function
for a Poisson random variable is

where p = mean of the probability function (and also the variance)
The cumulative probability function is

Univariate Analysis
For Multivariate Analysis, see McCuen, Reference 23, or other statistical texts.
The first step in data analysis is the selection of the best fitting probability
function, often beginning with a graphical analysis of the frequency histogram.
Moment ratios and moment-ratio diagrams (with 0, as abscissa and p, as ordinate)
are useful since probability functions of known distributions have characteristic
values of p, and p,.
Frequency analyszs is an alternative to moment-ratio analysis in selecting a
representative function. Probability paper (see Figure 1-59 for an example) is
available for each distribution, and the function is presented as a cumulative
probability function. If the data sample has the same distribution function as
the function used to scale the paper, the data will plot as a straight line.
The procedure is to fit the population frequency curve as a straight line using
the sample moments and parameters of the proposed probability function. The
data are then plotted by ordering the data from the largest event to the smallest
and using the rank (i) of the event to obtain a probability plotting position.
Two of the more common formulas are Weibull
pp, = i/(n + 1)