Probability Theory
Introduction
Statistics plays a fundamental role in decision making in many fields. For example, statistics is widely
used in the fields of Biology, Chemistry, Geology, Geography, etc, where statistics analysis aid in
informed decisions based on data. In this course we will discuss the use of statistical methods that
are often applied in the natural sciences. It is important to note that different techniques have
different assumptions and limitations.
Statistics as a study is concerned with methods for collecting, organising, summarising, presenting
and analysing data and making valid conclusions about the characteristics of the data. Statistics is
defined as “the scientific application of mathematical principles to the collection, analysis, and
presentation of numerical data” (http://www.amstat.org/careers/ whatisstatistics.cfm). Others
define it “as a discipline in the development and application of methods to collect, analyze and
interpret data” (http://statistics.unl.edu/whatis.shtml). Informed decision making is the role of a
branch of statistics called inferential statistics.
Inferential statistics are concerned with the interpretation and evaluation of data and making
statements about larger populations. Statistical inference is closely related to probability. The
development of the theories of probability has resulted in an increased scope of statistical
applications. Probability can be considered as the “essential thread” which runs throughout all
statistical inference, as it involves the study of uncertainty. The purpose of this chapter is to discuss
the nature of probability theory, which is the study of mathematical models of random phenomena.
Probability theory is the study of random phenomena, in the sense that it is concerned with those
properties of random phenomena that depend essentially on the notion of randomness and not on
any other aspects of the phenomenon considered.
Definitions
Population: Is a collection of all items or individuals under investigation. Individuals should have the
same characteristics and exist in the same controlled conditions or environment. There is however
natural variation among members in the population.
Parameter: A numerical description of some characteristic of the population. For example, the
population mean (μ) or the population variance (σ2).
Experiment: it is a process by which a measurement is taken or observations are made
generating outcomes. There are two types of experiments:
• Deterministic: This experiment result in the same outcomes being observed each time the
experiment is performed.
• Random: One of the several possible outcomes is observed each time an experiment is
performed. Statistical experiments are random experiments.
Outcome: it is any observable phenomenon, and it must be unique.
Sample space: its the collection of all possible outcomes of an experiment. It is denoted by S. A
sample space is also known as an event space, possibility space or simply the space. Note that the
word space has neither dimension nor volume; rather, a space is a complete collection of objects.
Introduction
Statistics plays a fundamental role in decision making in many fields. For example, statistics is widely
used in the fields of Biology, Chemistry, Geology, Geography, etc, where statistics analysis aid in
informed decisions based on data. In this course we will discuss the use of statistical methods that
are often applied in the natural sciences. It is important to note that different techniques have
different assumptions and limitations.
Statistics as a study is concerned with methods for collecting, organising, summarising, presenting
and analysing data and making valid conclusions about the characteristics of the data. Statistics is
defined as “the scientific application of mathematical principles to the collection, analysis, and
presentation of numerical data” (http://www.amstat.org/careers/ whatisstatistics.cfm). Others
define it “as a discipline in the development and application of methods to collect, analyze and
interpret data” (http://statistics.unl.edu/whatis.shtml). Informed decision making is the role of a
branch of statistics called inferential statistics.
Inferential statistics are concerned with the interpretation and evaluation of data and making
statements about larger populations. Statistical inference is closely related to probability. The
development of the theories of probability has resulted in an increased scope of statistical
applications. Probability can be considered as the “essential thread” which runs throughout all
statistical inference, as it involves the study of uncertainty. The purpose of this chapter is to discuss
the nature of probability theory, which is the study of mathematical models of random phenomena.
Probability theory is the study of random phenomena, in the sense that it is concerned with those
properties of random phenomena that depend essentially on the notion of randomness and not on
any other aspects of the phenomenon considered.
Definitions
Population: Is a collection of all items or individuals under investigation. Individuals should have the
same characteristics and exist in the same controlled conditions or environment. There is however
natural variation among members in the population.
Parameter: A numerical description of some characteristic of the population. For example, the
population mean (μ) or the population variance (σ2).
Experiment: it is a process by which a measurement is taken or observations are made
generating outcomes. There are two types of experiments:
• Deterministic: This experiment result in the same outcomes being observed each time the
experiment is performed.
• Random: One of the several possible outcomes is observed each time an experiment is
performed. Statistical experiments are random experiments.
Outcome: it is any observable phenomenon, and it must be unique.
Sample space: its the collection of all possible outcomes of an experiment. It is denoted by S. A
sample space is also known as an event space, possibility space or simply the space. Note that the
word space has neither dimension nor volume; rather, a space is a complete collection of objects.
