23
C H A P T E R 2
Fundamental Reliability
Mathematics
2.1 INTRODUCTION
is chapter will first briefly discuss fundamental concepts of set theory and probability, then the
probability density function (PDF) and the cumulative distribution function (CDF) of random
variables, and finally typical types of distributions and the goodness-of-fit test. e focus of
these discussions is not to explain these in a mathematical approach, but how to implement
the probabilistic concepts in the reliability-based mechanical design for design engineers. e
calculation of the PDF and the CDF will be mainly through Excel or MATLAB
®
.
2.2 EXPERIMENT, OUTCOME, SAMPLE SPACE, AND
EVENT
Every physical parameter in engineering design or real life has a certain element of uncertainty.
For example, a wind speed against a building, the weight of a lifted object of a bridge-crane,
and the diameter of a machined shaft cannot be predicted exactly. e terms of experiment,
outcome, sample space, and the event can be used to describe these phenomena.
Experiment refers to an act of doing something, but the results of it cannot be predicted exactly
before the action has been completed. For example, conducting a material tension test is an
experiment. Before the tensile test is completed, the actual yield strength of a test specimen is
not known. Conduction of 2 plus 2 is not an experiment, because the result of it can be predicted.
For another example, tossing a coin or rolling dice is an experiment.
Sample point or outcome refers to the single result of an experiment. For example, rolling a die
will have six possible outcomes, which are 1, 2, 3, 4, 5, and 6. For tossing a coin, the occurrence
of “head” is a sample point or an outcome.
Per the definition of an experiment, an experiment must have at least two outcomes
but also can have an infinite number of possible outcomes. For example, tossing a coin will
only have two outcomes: head or tail. e diameter of a qualified shaft with a dimension
;1:000 ˙ 0:005
00
will have infinite possible outcomes, in which the value of the diameter can
be any numerical value between 0.995
00
and 1.005
00
.
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