Page 89 - Using ANSYS for Finite Element Analysis Dynamic, Probabilistic, Design and Heat Transfer Analysis

76 • using ansys for finite eLement anaLysis
• For univariate data, it is often useful to determine a reasonable dis-
tributional model for the data.
• Statistical intervals and hypothesis tests are often based on spe-
cific distributional assumptions. Before computing an interval or
test based on a distributional assumption, we need to verify that
the assumption is justified for the given data set. In this case, the
distribution does not need to be the best-fitting distribution for the
data, but an adequate enough model so that the statistical technique
yields valid conclusions.
• Simulation studies with random numbers generated from using a
specific probability distribution are often needed.

The mathematical definition of a continuous probability function,
f(x), is a function that satisfies the following properties.

1. The probability that x is between two points a and b is:

[
()
pa ≤ x ≤ b] = ∫ a b fx dx
2. It is non-negative for all real x.
3. The integral of the probability function is one, that is:

∞
()
∫ −∞ fx dx = 1

What does this actually mean? Since continuous probability
functions are defined for an infinite number of points over a continuous
interval, the probability at a single point is always zero. Probabilities are
measured over intervals, not single points. That is, the area under the
curve between two distinct points defines the probability for that inter-
val. This means that the height of the probability function can in fact be
greater than one. The property that the integral must equal one is equiva-
lent to the property for discrete distributions that the sum of all the prob-
abilities must equal one.
Probability distributions are typically defined in terms of the proba-
bility density function (pdf). However, there are a number of probability
functions used in applications. For a continuous function, the pdf is the
probability that the variate has the value x. Since for continuous distri-
butions the probability at a single point is zero, this is often expressed in
terms of an integral between two points.