www.soran.edu.iq Probability and Statistics Dr. Saeid Moloudzadeh Sample Space and Events 1 Contents • Descriptive Statistics • Axioms of Probability • Combinatorial Methods • Conditional Probability and Independence • Distribution Functions and Discrete Random Variables • Special Discrete Distributions • Continuous Random Variables • Special Continuous Distributions • Bivariate Distributions
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Www.soran.edu.iq Probability and Statistics Dr. Saeid Moloudzadeh Sample Space and Events 1 Contents Descriptive Statistics Axioms of Probability Combinatorial.
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www.soran.edu.iq 1
Probability and Statistics
Dr. Saeid Moloudzadeh
Sample Space and Events
Contents• Descriptive Statistics• Axioms of Probability• Combinatorial Methods • Conditional Probability and
Independence • Distribution Functions and
Discrete Random Variables• Special Discrete Distributions • Continuous Random Variables • Special Continuous Distributions • Bivariate Distributions
Contents• Descriptive Statistics• Axioms of Probability• Combinatorial Methods • Conditional Probability and Independence • Distribution Functions and Discrete Random Variables• Special Discrete Distributions • Continuous Random Variables • Special Continuous Distributions • Bivariate Distributions
Definition (Experiment): An experiment is an act or process of observation that leads to a single outcome that cannot be predicted with certainty.
A sample point is the most basic outcome of an experiment.
Definition (Sample space): The set of all possible outcomes to a random experiment (all the sample points) is called the sample space and is denoted by S.
In the study of probability theory the relations between different events of an experiment play a central role. In the remainder of this section we study these relations. In all of the following definitions the events belong to a fixed sample space S.
Definition:• Subset: An event A is said to be a subset of the event B if,
whenever A occurs, B also occurs. This means that all of the sample points of A are contained in B. Hence considering A and B solely as two sets, A is a subset of B in the usual set-theoretic sense: that is,
Equality: Events A and B are said to be equal if the occurrence of A implies the occurrence of B, and vice versa; that is, if and hence A=B.
Intersection: An event is called the intersection of two events A and B if it occurs only whenever A and B occur simultaneously. In the language of sets this event is denoted by AB or because it is the set containing exactly the common points of A and B.
• Union: An event is called the union of two events A and B if it occurs whenever at least one of them occurs. This event is since all of its points are in A or B or both.
• Complement: An event is called the complement of the event A if it only occurs whenever A does not occur. The complement of A is denoted by
• Difference: An event is called the difference of two events A and B if it occurs whenever A occurs but B does not. The difference of the events A and B is denoted by A-B.
• Certainty: An event is called certain if its occurrence is inevitable. Thus the sample space is a certain event.
• Impossibility: An event is called impossible if there is certainty in its nonoccurrence.
Therefore, the empty set is an impossible event.• Mutually Exclusiveness : If the joint occurrence of two
events A and B is impossible, we say that A and B are mutually exclusive or disjoint. Thus A and B are mutually exclusive if the occurrence of A precludes the occurrence of B, and vice versa. Since the event representing the joint occurrence of A and B is AB, their intersection, A and B, are mutually exclusive if AB =
Example 2.3: At a busy international airport, arriving planes land on a first-come, first-served basis. Let
E = there are at least five planes waiting to land,F = there are at most three planes waiting to land,H = there are exactly two planes waiting to land.Then