Basic Mathematical Concepts

Introduction

Just like in the development of other areas of science and technology, mathematics has played an important role in the development of the reliability field also. Although the origin of the word “mathematics” may be traced back to the ancient Greek word “mathema”, which means “science, knowledge, or learning”, the history of our current number symbols, sometimes referred to as the “Hindu-Arabic numeral system” in the published literature [1], goes back the third century вс [1 ]. Among the early evidences of these number symbols’ use are notches found on stone columns erected by the Scythian Emperor of India named Asoka, in around 250 ВС [ 1].

The history of probability goes back to the gambling manual written by Girolamo Cardano (1501-1576), in which he considered some interesting issues concerning probability [1,2]. However, Pierre Fermat (1601-1665) and Blaise Pascal (1623— 1662) were the first two individuals who independently and correctly solved the problem of dividing the winnings in a game of chance [1,2]. Pierre Fermat also introduced the idea of “differentiation”. In modern probability theory, Boolean algebra plays a pivotal role and is named after an English mathematician George Boole (1815-1864), who published, in 1847, a pamphlet titled The Mathematical Analysis of Logic: Being an Essay towards a Calculus of Deductive Reasoning [1-3].

Laplace transforms, often used in reliability area for finding solutions to first-order differential equations, were developed by a French mathematician named Pierre- Simon Laplace (1749-1827). Additional information on mathematics and probability history is available in Refs. [1,2].

This chapter presents basic mathematical concepts that will be useful to understand subsequent chapters of this book.

Arithmetic Mean and Mean Deviation

A set of given reliability data is useful only if it is analyzed properly. More specifically, there are certain characteristics of the data that are useful for describing the nature of a given data set, thus enabling better decisions related to the data. This section presents two statistical measures considered useful for studying engineering system reliability-related data [4,5].

Arithmetic Mean

Often, the arithmetic mean is simply referred to as mean and is expressed by where

m is the mean value (i.e., arithmetic mean). к is the number of data values.

Xj is the data value i, for i = 1,2,3,

Example 2.1

Assume that the quality control department of a company involved in the manufacture of systems/equipment for use in mines inspected six identical systems/equipment and found 2, 1, 5, 4, 7, and 3 defects in each system/ equipment. Calculate the average number of defects per system/equipment (i.e., arithmetic mean).

By substituting the given data values into Equation (2.1), we get

Thus, the average number of defects per system/equipment is 3.6. In other words, the arithmetic mean of the given dataset is 3.6.

Mean Deviation

This is a measure of dispersion whose value indicates the degree to which given data set tends to spread about a mean value. Mean deviation is expressed by

where

MD is the mean deviation. к is the number of data values.

D, is the data value i, for i = 1,2,3, k.

|D, -m is the absolute value of the deviation of Д from m.

Example 2.2

Calculate the mean deviation of the dataset provided in Example 2.1. Using the Example 2.1 dataset and its calculated mean value (i.e., m = 3.6 defects per system/equipment) in Equation (2.2), we obtain

Thus, the mean deviation of the dataset provided in Example 2.1 is 1.6.

Boolean Algebra Laws

Boolean algebra plays an important role in various types of reliability studies and is named after George Boole (1813-1864), an English mathematician. Five of the Boolean algebra laws are as follows [3,6]:

• Law I: Commutative Law:

where

Y is an arbitrary set or event.

X is an arbitrary set or event.

+ denotes the union of sets.

where

Dot (.) denotes the intersection of sets. Note that Equation (2.4) sometimes is written without the dot (e.g., YX), but it still conveys the same meaning.

• Law II: Idempotent Law:

• Law III: Associative Law:

where

Z is an arbitrary set or event.

• Law IV: Distributive Law:

• Law V: Absorption Law:

Probability Definition and Properties

Probability is defined as follows [7]:

where

P(Z) is the probability of occurrence of event Z.

N is the number of times event Z occurs in the n repeated experiments.

Some of the basic probability properties are as follows [7,8]:

• The probability of occurrence of event, say Z, is

• The probability of the occurrence and nonoccurrence of an event, say event Z, is always:

where

P(Z) is the probability of occurrence of event Z.

P(Z) is the probability of nonoccurrence of event Z.

• The probability of the union of n independent events is where

P(Zt) is the probability of occurrence of event Z(, for /=1,2, 3,..., n.

• The probability of the union of n mutually exclusive events is

• The probability of an intersection of n independent events is

Example 2.3

Assume that a transportation system has two critical subsystems Zx and Z2. The failure of either subsystem can, directly or indirectly, result in an accident. The failure probability of subsystems Zx and Z2 is 0.08 and 0.05, respectively.

Calculate the probability of occurrence of the transportation system accident if both of these subsystems fail independently.

By substituting the given data values into Equation (2.16), we get

Thus, the probability of occurrence of the transportation system accident is 0.126.

 
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