introduction_to_probability

introduction_to_probability - Dr. Michael Monticino...

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Dr. Michael Monticino niversity of North Texas University of North Texas Department of Mathematics
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± One of central advances of the 20 th century was the understanding, quantification and mastery of uncertainty ± Without the tools of probability and statistics, many of the advances in engineering , y gg and science might never have happened
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± Discrete probability ± Random variables
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± Probability framework ± Basic axioms and consequences ± Equally likely sample spaces ± Counting techniques ± Conditional probability ± Bayes’Theorem ± Independence
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± Probability is a framework for quantifying and reasoning about uncertainty ± Alternate viewpoints about what are appropriate applications and the correct interpretation of the probability concepts ± Frequency ± Subjective ± Logical
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± Frequency viewpoint ± The probability of an event is interpreted as the long run proportion of times that the vent occurs when repeating an event occurs when repeating an experiment a large number of times ± For a frequentist ,probability has no meaning other than to express this relative frequency
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± Subjectivist or Bayesian viewpoint interprets probability as the degree of elief n individual has in what the belief an individual has in what the outcome of an experiment ± Probability involves a subjective level f knowledge about an event and of knowledge about an event and applies to circumstances other than those that can be represented as repeatable experiments ± Allows use of prior knowledge when pdating probabilities given updating probabilities given additional information
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± Logical or Objective view is that two persons with the same information would arrive at the same probability of an event robabilities are not relative to ± Probabilities are not relative to individual, but to the situation ± Generally agree that probability constructs apply to circumstances other than ose that can be represented those that can be represented as repeatable experiments
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th th i th f i ± Other than in the use of prior information, there is little practical consequence on the use of probability from the differing philosophical frameworks on how probability is terpreted interpreted ± Axioms of probability are the same
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sample space s the collection outcomes ± A sample space, , is the collection outcomes from an experiment ± n event A is a subset of the sample space Ω An event, A, is a subset of the sample space ± Probabilities are assigned to events, such that g , ± 1 ) ( = Ω P ± , for any event A ± If A 1 , A 2 , ... are disjoint events, then 01 ≤≤ P A () = = = 1 1 ) ( ) ( n n n n A P A P U
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± P(A c ) = 1 P(A) ± P ( ) ∅ = 0 ± If ,then A B P A P B ( ) ( ) ± ) ( ) ( ) ( ) ( B A P B P A P B A P + =
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± A coin is tossed 3 times with the results (head 3 ( or tail) recorded for each toss. Assume that each possible sequence of heads/tails has probability 1/8. What is the probability of getting at least one head?
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This note was uploaded on 10/28/2009 for the course MATH ma024 taught by Professor Thu during the Spring '09 term at Vienna EBA.

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introduction_to_probability - Dr. Michael Monticino...

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