PROBABILITY AND STATISTICS FOR ENGINEERS

1. In The Name of God Malayer University Department of Civil Engineering PROBABILITY AND STATISTICS FOR ENGINEERS Taught by: Dr. Ali Reza Bagherieh

2. Introduction This course is concerned with the development of basic principles in constructing probability models and the subsequent analysis of these models.

3. Probability and Random Variables Basic Probability Concepts In studying a random phenomenon, we are dealing with an experiment of which the outcome is not predictable in advance. Example: Games of chance Physical or natural phenomena involving uncertainties Uncertainty comes from : 1-complexity 2-lack of understanding of all the causes and effects 3-lack of information For example, weather prediction The second class of problems widely studied by means of probabilistic models concerns those exhibiting variability. for example, a problem in traffic flow where an engineer wishes to know the number of vehicles crossing a certain point on a road within a specified interval of time.

4. ELEMENTS OF SET THEORY A set is a collection of objects possessing some common properties These objects are called elements of the set and they can be of any kind with any specified properties. A set containing no elements is called an empty or null set and is denoted by Φ finite sets ------infinite sets

5. enumerable or countable set All of its elements can be arranged in such a way that there is a one-to-one correspondence between them and all positive integers nonenumerable or uncountable If every element of a set A is also an element of a set B, the set A is called a subset of B and this is represented symbolically by

6. The ‘largest’ set containing all elements of all the sets under consideration is called space and is denoted by the symbol S.

7. If AB =Φ, sets A and B contain no common elements, and we call A and B disjoint.

9. S AM P L E S P A CE AN D P RO BA BILIT Y M E AS U RE In probability theory, we are concerned with an experiment with an outcome depending on chance, which is called a random experiment. It is assumed that all possible distinct outcomes of a random experiment are known and that they are elements of a fundamental set known as the sample space. Each possible outcome is called a sample point, An event is generally referred to as a subset of the sample space having one or more sample points as its elements. For a given random experiment, the associated sample space is not unique Its construction depends upon the point of view adopted as well as the questions to be answered.

10. All relations between outcomes or events in probability theory can be described by sets and set operations.

11. Disjoint sets are mutually exclusive events in probability theory.

12. Given a random experiment, a finite number P(A) is assigned to every event A in the sample space S of all possible events. The number P(A) is a function of set A and is assumed to be defined for all sets in S. It is thus a set function, and P(A) is called the probability measure of A or simply the probability of A.

16. ASSIGNMENT OF PROBABILITY For problems in applied sciences, a natural way to assign the probability of an event is through the observation of relative frequency. Assuming that a random experiment is performed a large number of times, say n, then for any event A let nA be the number of occurrences of A in the n trials and define the ratio nA /n as the relative frequency of A. Under stable or statistical regularity conditions, it is expected that this ratio will tend to a unique limit as n becomes large. Another common but more subjective approach to probability assignment is that of relative likelihood. When it is not feasible or is impossible to perform an experiment a large number of times, the probability of an event may be assigned as a result of subjective judgment. The statement ‘there is a 40% probability of rain tomorrow’ is an example in this interpretation, where the number 0.4 is assigned on the basis of available information and professional judgment.

17. STATISTICAL INDEPENDENCE Given individual probabilities P(A) and P(B) of two events A and B, what is P(AB)? A special case in which the occurrence or nonoccurrence of one does not affect the occurrence or nonoccurrence of the other. In this situation events A and B are called statistically independent or simply independent

22. CONDITIONAL PROBABILITY Given two events A and B associated with a random experiment, P(A/B) probability is defined as the conditional probability of A, given that B has occurred.

24. Conditional probabilities are probabilities They satisfy the probability axioms.

25. As for the third axiom, if A1, A2, . . . are mutually exclusive, then A1B, A2B, . . . are also mutually exclusive. Hence,

27. Theorem 2. 1: theorem of total probability . Suppose that events B1, B2, . . . , and Bn are mutually exclusive and exhaustive (i.e. S B1 B2 Bn). Then, for an arbitrary event A,

32. Random Variables and Probability Distributions We can make statements concerning the events that can occur, and these statements are made based on probabilities assigned to simple outcomes. A systematic and unified procedure is needed to facilitate making these statements Each of the possible outcomes of a random experiment be represented by a real number. Each outcome is identified by its assigned real number rather than by its physical description. A sample space of arbitrary elements by a new sample space having only real numbers as its elements Use arithmetic means for probability calculations.

33. RANDOM VARIABLES We assume that it is possible to assign a real number X(s) for each outcome s following a certain set of rules. The ‘number’ X(s) is really a real-valued point function defined over the domain of the basic probability space

34. Let us again assign number one to the event success and zero to failure. If X is the random variable associated with this experiment, then takes on two possible values: 1 and 0. The random variable X is called a random variable if it is defined over a sample space having a finite or a countably infinite number of sample points. In this case, random variable takes on discrete values, and it is possible to enumerate all the values it may assume. In the case of a sample space having an uncountably infinite number of sample points, the associated random variable is called a continuous random variable, with its values distributed over one or more continuous intervals on the real line. We make this distinction because they require different probability assignment considerations.

35. We note here that an analysis involving random variables is equivalent to considering a random vector having the random variables as its components. PROBABILITY DISTRIBUTIONS The behavior of a random variable is characterized by its probability distribution, that is, by the way probabilities are distributed over the values it assumes. A probability distribution function and a probability mass function are two ways to characterize this distribution for a discrete random variable. They are equivalent in the sense that the knowledge of either one completely specifies the random variable.

36. Probability Distribution Function

37. Important properties possessed by a PDF

41. completely characterizes random variable

44. PROBABILITY DENSITY FUNCTION FOR CONTINUOUS RANDOM VARIABLES