GE330_lect19 - Lecture 19 Probability Review April 15, 2009...

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Lecture 19 Probability Review April 15, 2009
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Basic Review Basic definitions of probability theory Addition law, Conditional probability, Bayes rule Random variables and probability distributions Four important probability distributions The material is in Chapter 12. 1
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Basic Definitions The motivation of study probability is to provide mathemat- ical models for nondeterministic situations. An experiment refers to the process of obtaining an ob- served result of some phenomenon. A performance of an experiment is call a trial and an ob- served result is called an outcome . The set of all possible outcomes is referred to as the sample space . A subset of the sample space is called as an event . We say that an event occurs if it contains the outcome that occurred. An event is called an elementary event if it contains exactly one outcome of the experiment. 2
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Examples An experiment consists of tossing two coins, and the ob- served face of each coin is of interest. The set of possible outcomes (sample space) is: S = { HH,HT,TH,TT } The event of obtaining “at least one head” corresponds to the subset A = { HH,HT,TH } . The event with A = { HH } is an elementary event. If a coin is tolled repeatedly until a head occurs, the natural sample space is S = { H,TH,TTH, ···} . The event of obtaining “a head in less than 4 tosses” corre- sponds to A = { H,TH,TTH,TTTH } . A light bulb is placed in service and the time of operation un- til it burns out is measured. Conceptually, the sample space can be S = { t | 0 t < ∞} . The event that the bulb burns out in T units of time corre- sponds to A = { t | 0 t T } . 3
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Probabilities Probability and relative frequency. If we repeat an experiment n times, and an event occurs m ( A ) times. Then the relative frequency is given by m ( A ) n and the probability of event E , P { E } is: P { E } = lim n →∞ m n . The probability is always a number between 0 and 1, i.e., for an event E , 0 P { E } ≤ 1 . Let S be the whole sample space and be the empty set, we always have P { S } = 1 and P {∅} = 0. One can also regard probability as a subjective measure of belief that an event will occur.
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This note was uploaded on 08/31/2010 for the course IESE GE 330 taught by Professor Nedich during the Spring '09 term at University of Illinois at Urbana–Champaign.

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GE330_lect19 - Lecture 19 Probability Review April 15, 2009...

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