Lecture V
Probability
theory
Lecture questions
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Classical definition of probability
Frequency probability
Discrete variable and probability distribution
Continuous variable and probability density
function
Probability theory
Probability theory is the branch of mathematics
concerned with analysis of random phenomena. The
central objects of probability theory are random
variables, stochastic processes (random processes),
and events. The probability theory is a mathematical
foundation for statistics. As a mathematical foundation
for statistics, probability theory is essential to many
human activities that involve quantitative analysis of
large sets of data. Methods of probability theory also
apply to descriptions of complex systems given only
Basic Concepts
• Random Experiment. An experiment is said to be a
random experiment, if it's out-come can't be predicted
with certainty.
• Sample Space. The set of all possible out-comes of an
experiment is called the sample space. The elements of
the sample space are called sample point or event point.
• Event. Every subset of a sample space is an event.
• Simple event. An event, consisting of a single sample
point is called a simple event.
• Compound event. A subset of the sample space, which
has more than one element is called a mixed event.
• Exhaustive events. When every possible outcome of an
experiment is considered.
Classical definition of probability
• The probability of an event is the ratio of the
number of cases m favorable for the event to the
number of total outcomes n(s) possible in an
equiprobable sample space.
m( E )
p( E ) 
n( s )
• Note: This definition is not true, if
(a) The events are not equiprobable.
(b) The possible outcomes are infinite.
Statistical probability (frequency
probability)
• Frequency (absolute frequency) is the number
of times an event actually occurs in any
experiment.
• The relative frequency of an event is the ratio
of the absolute frequency of a particular
m( E ) the total number of
successful event,
with
*
p (E) 
n
events.
• Statistical probability (frequency
probability)
m( E )
*
p ( E )of lim
p ( E )frequency
 lim
defines as the limit
relative
n
n 
n of an
event in a large number of trials.
Discrete and continuous variables
• A random variable (or stochastic variable) is a
way of assigning a value (often a real number to
each possible outcome of a random event.
• A discrete random variable maps events to
values of a countable set (e.g., the integers, with
each value in the range having probability
greater than or equal to zero.
• A continuous random variable maps events to
values of an uncountable set (e.g., the real
numbers). For a continuous random variable,
the probability of any specific value is zero,
whereas the probability of some infinite set of
Probability distribution
• A probability distribution identifies the
probability of each value of a random
variable when the variable is discrete and
the probability of the value falling within a
particular interval when the variable is
continuous. The probability distribution of
a continuous random variable is
completely described by probability
density function.
Moments
The probability distribution of a random
variable is often characterized by a small
number of parameters, which also have a
practical interpretation. They are
• The expected value
• The variance
• Standard deviation
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