M2L1. Random Events and Probability Concept


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1 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.
2 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.
3 5. Events Event is defined as a subset of the Sample Space. Simple or Elementary Event consists only one single element of Sample Space. Examples of Sample Space and Events are: Fig. 2.Events 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
4 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 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/mm 2 ), 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.
5 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 nonnegative 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 occur simultaneously. defined as an outcome when or or both 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.
6 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 random in the time interval, the probability that it may occur in the interval, 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 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,. There are two possibilities: If 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:
7 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. McGrawHillScience/Engineering/Math. ISBN:
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