Download M2L1 Random Events and Probability Concept

M2L1
Random Events and Probability Concept
1. Introduction
In this lecture, discussion on various basic properties of random variables and definitions of
different terms used in probability theory and its concept are explained.
2. Random Experiment
Understanding on random experiment is required to define ‘experiment’ first. In logic,
‘experiment’ is a set of conditions under which behavior of some variables are observed.
Random experiment is an experiment, under some conditions, in which the outcome can not be
predicted with certainty. In other words, ‘random experiment’ is the ‘experiment’ where it is not
possible to ascertain or control the values of certain variables, and results vary from one
performance/trial to the other. For example, tossing coin(s), rolling dice(s) etc.
To run a random experiment, it is important to decide on choosing the input samples. There are
two sampling techniques commonly used:
a. With Replacement – where each item in the chosen set of variables for a particular
experiment or ‘sample space’ is replaced before drawing the next sample,
b. Without replacement – where samples are drawn without replacement in the sample
space
Random experiments, based on mutual dependence among different set of results, can be of two
types:
a. Independent experiment – where the outcome of one experiment has no influence on
any of such other experiments/trials. e.g. successive tossing of a coin.
b. Compound experiment – where a new experiment is carried out by performing any ‘n’
number of experiments in sequence. e.g., tossing of two coins one after another and
getting ‘head’ for both.
The Independent experiment is also called as Simple experiment.
3. Trials
Simple experiment or trials can be classified based on number of possible outcomes as follows:
a. Bernoulli’s trials – when a simple experiment has only two outcomes (either success or
failure) resulted from independent replication of experiment.
b. Multinomial trials – when a simple experiment has ‘k’ number of outcomes resulted
from independent replication of experiment.
4. Sample Space
This is formally defined as follows:
A set ‘S’ that consists of all possible outcomes of a random experiment is called a Sample Space,
and each outcome is called a Sample Point.
e.g. if a dice is tossed, the sample space, or set of all possible outcomes, is given by 1, 2, …, 6.
Fig. 1. Sample Space
Sample space can be classified either as Discrete or Continuous Sample Space.
A Sample Space is discrete if all subsets correspond to events. Whereas, a Sample Space is
continuous, when only special subset or a ‘measurable’ subset, on which a probability can be
assigned (also know as   algebra), corresponds to events.
5. Events
Event is defined as a subset of the Sample Space. Simple or Elementary Event consists only one
single element of Sample Space.
Fig. 2.Events
Examples of Sample Space and Events are:
i.
In case of reservoir storage, the range of levels from zero to maximum capacity forms the
sample space. Typical example of events in this case may be ‘reservoir storage above
dead storage’, ‘below 50% of total capacity’ etc.
ii.
In case of ‘traffic volume’, does ‘all possible types of vehicle’ constitute the sample
space?
Answer is NO. The traffic volume means the total ‘number’ of different types of vehicles
moving across a particular stretch of a road. Thus, the sample space for ‘traffic volume’
consists of number of each type of vehicle, which varies between zero and infinity. In this
case, any range of real integer numbers can be treated as an event.
In some other ‘experiment’, ‘all possible types of vehicle’ can also treated as an sample
space. Reader may find it out oneself using content of this chapter.
6. Concept of Probability
The probability concept was proposed originally to explain the uncertainty involved in outcome
of random events. In this theory, it is considered that random events can occur either sequentially
or simultaneously, e.g. occurrence of road accidents in transportation safety analysis, occurrence
of extreme (either very high or very low) rainfall, which is beyond the capacity of drainage
network etc.
Generally, for a particular system, occurrence of such events approaches to a proportion of total
number of events. With the increase in number of observations, this proportion becomes
constant. For example, tossing of a coin is an ‘experiment’ and noting the outcome. Here
estimation of the percentage of heads (or tail) approaches to 0.5 with increase in number of
tossing (for a fair coin). This gives rise to the relative frequency approach for interpretation of
probability.
7. Interpretation of Probability
Probability can be interpreted in various ways. Based on relative frequency approach we can say,
if an experiment is performed ‘ ’ times and an event ‘ ’ occurs
times, then with high degree
of certainty, the relative frequency
is close to the probability of occurrence of ‘ ’,
(Papoulis and Pillai, 2002):
This is true when ‘ ’is sufficiently large, that is,
8. Assigning Probability
There are basically three ways to assign probability to the results of an event.
Firstly, probabilities of certain events
in an inexact way, e.g. if out of
days in a year,
65 days have recorded rainfall above the average; then the probability of the event: ‘above
average daily rainfall’ is assigned as 65/365 or 0.178.
Secondly, make an analytical reasoning for the event for which probability is to be assigned, e.g.
as per the definition of characteristic strength of concrete, if a
cm concrete cube is made of
grade concrete and subject to a pressure of
the cube will fail, is
MPa (N/mm2), probability of the event, that
.
Third and lastly, assume that the probability of an event will follow certain axioms and then
using deductive approach calculate the probability of an event based on probabilities of other
related events, e.g. the probability that a testing device will be rated as ‘very poor’, ‘poor’,
‘average’, ‘satisfactory’ and ‘excellent’ are
,
,
,
and
respectively; then the
probability of the same device will be rated as ‘above average’ will be
.
The most important point to remember, while assigning probability, is that ‘probability’ cannot
be assigned to an ‘experiment’, rather to the ‘results of an experiment’ or ‘event’.
9. Axiomatic Definitions
If there is set, ‘ ’ of mutually exclusive events i.e. one event exclude occurrence of the other
events, then
Axiom1: Probability of ,
is an non-negative number:
Axiom 2: Total probability of all events in the set, ‘ ’ is:
Axiom 3: If ‘A’ and ‘B’ are two mutually exclusive events belong to S, the probability of event
in addition to individual probabilities of and :
where, union of two events ‘ ’ and ‘ ’ or
defined as an outcome when ‘ ’ or ‘ ’ or both
occur simultaneously.
10. Classical Definition
The probability of an event ‘ ’,
is determined, without actual random experiment, as a
ratio of favourable and all possible outcomes of an experiment. In terms of mathematical
expression:
where, ‘
’ is the favourable related to event ‘ ’ and ‘ ’ is the total possible outcomes.
It should be noted that, this definition of probability implicitly assumes that all possible
outcomes of an experiment are equally likely.
To understand the classical definition, there are a few critical views one has to keep in mind:
a. the term ‘equally likely’ means that outcomes of an event are equally probable or fair
choice which may not be feasible and determination of
,
is difficult in practical
cases.
b. this definition is applicable to limited practical problems, as ‘equal probability of
choices’ is hard to achieve.
c. If the number of possible outcome is infinity, some sort of measurement of infinity,
such as, length, area etc. should be assigned to get the ratio of
and
.
The classical definition of probability is valid for following cases:
i.
In applications where the assumption of having ‘equally likely’ outcomes can be
established through long experiment size, e.g. if the occurrence of cyclonic storm is
time interval, the probability that it may occur in the interval
random in the
, equals to
ii.
.
In applications where it is impossible to repeat an experiment for sufficiently large
number of times, it is assumed that results are equally likely.
11. Determination of Probability
To determine probability of an event one has to follow the three steps described below:
1. Assign the probabilities
for favourable events in an inexact way
2. Check whether all the axioms are followed and then calculate the probability of the
favourable events using deductive approach
3. Make a physical guess for events based on probabilities of events already calculated e.g.
.
12. Probability and Occurrence
Chance of occurrence of an event, ‘ ’ for a particular trial or run of an experiment can be
determined based on size of its probability,
If
. There are two possibilities:
is small, such as, 0.03, then it concludes only a certain ‘degree of confidence’ that event
‘ ’ will occur.
If
is closer to 1, such as, 0.999, then it concludes that with practical certainty, ‘ ’ will
occur in the next trial of the experiment.
However, these possibilities are ‘measure of belief’ or subjective in nature, cannot be tested
experimentally.
13. Randomness and Causation
Randomness and Causation are two properties of outcomes of an experiment or occurrence of
any event. There is prominent contrasts between these two properties:
a. Randomness links with probabilistic or stochastic system, whereas Causation links with
deterministic system.
b. Randomness is always defined with certain errors and range of relevant parameters,
whereas Causation is defined with a high degree of certainty if number of outcomes is
large enough.
14. Concluding Remarks
Before closing this lecture there are few important points to remember in brief. Random events
are possible outcomes of a random experiment, and probability is a measure of uncertainty in
occurrence of them. Random events consist of either single point or multiple outcomes in a
sample space. The relationships among random events are governed by the Set Theories and
event properties. These properties will be explained in the next lecture.
References:
Papoulis, A. and Pillai, S. U. (2002). Probability, Random Variables and Stochastic Processes.
Fourth Edition. McGraw-HillScience/Engineering/Math. ISBN: 978-0072817256.