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a
b
Fig. 3.11 a Probability density function f ( x ), and b cumulative distribution function F ( x ), of a
ˇ2 distribution with dif erent values for ʦ.
h e probability density function of the ˇ 2 distribution is
where x >0, otherwise f ( x )=0; ʓ is again the Gamma function. Once again, ʦ
is the number of degrees of freedom (Fig. 3.11).
3.5 Examples of Theoretical Distributions
h e function randtool is a tool for simulating discrete data sets with statistics
similar to our data set. h is function creates a histogram of random numbers
from the distributions in the Statistics Toolbox (MathWorks 2014). h e
random numbers that have been generated by using this tool can then be
exported into the workspace. We start the graphical user interface (GUI) of
the function by typing
Movie
3.1
randtool
at er the prompt. We can now create a data set similar to the one in the i le
organicmatter_one.txt . h e 60 measurements have a mean of 12.3448 wt%
and a standard deviation of 1.1660 wt%. h e GUI uses Mu for ʼ (the mean
of a population) and Sigma for ˃ (the standard deviation). At er choosing
Normal for a Gaussian distribution and 60 for the number of samples, we get a
histogram similar to the one in the i rst example (Section 3.3). h is synthetic
distribution based on 60 measurements represents a rough estimate of the
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