Game Development Reference
In-Depth Information
Figure 15-3.
An inverse cumulative distribution function
Gaussian Distribution
Now that we've learned some general concepts about probability functions, let's learn about
some specific types of probability functions. One of the most commonly used probability functions
is the
Gaussian
or normal distribution. Gaussian distributions are widely used to model physical
effects for subjects ranging from the height of women in Europe to the energy states of molecules.
The Gaussian distribution is a symmetrical probability function. It is the “Bell curve” that
your teachers have been threatening you with all these years. The curve shown in Figure 15-1
is a Gaussian distribution. It is a symmetric probability function. The curve to the left of the
mean value has the same shape as the curve to the right of the mean value. The Gaussian distri-
bution function is called a normal distribution because the area under the probability function
curve is equal to 1. An exponential mathematical equation characterizes the Gaussian distribution.
2
(
x
−
μ
σ
)
1
−
2
fx
()
=
e
2
(15.2)
2
2
πσ
is the mean value of the function. As described in the previous section, it is
the peak of the probability function curve. The quantity
f(x)
is the probability for a given value
of
x
. The quantity
The quantity
μ
is called the
standard deviation
and is a measure of the narrowness or
thickness of the profile. As shown in Figure 15-4, a lower standard deviation,
σ
= 0.5 for example,
results in a narrower profile, meaning that most of the variable values will be close to the mean
value. A larger standard deviation indicates a wider profile, which means there is a greater
range of likely values.
σ
