MAT4371 Summary.docx

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University of Ottawa
Francois Theberge

Chapter 1Introduction Sample space the set of all possible outcomes of an experiment denoted as the set Example the sample space of flipping a coin is Alternate definition The sample spaceis a set which has a 1to1 correspondence with theoutcomes of a random experimentDiscretetime stochastic process suppose the random experiment with sample space if performed once per time unit We define the sample space of the composite experiment consisting of the sequenceof simple experiments as called a discretetime stochasticprocess Example an infinite number of coin flips algebra a collection F of subsets is aalgebra if 12 closed under complementation 3 countable union This is equivalent to sinceTotal information ifwe say we have total information If not we say we have partial informationEvent any subset of the sample space Example the event that the coin lands on heads then We say that the eventoccurs when the outcome of the experiment lies inUnion For any two events and consists of all the outcomes that are in or or both Adding the two sets together into a new set Intersection For any two events andconsists of all the outcomes that are in both and Take all the elements in common with the two sets and make a new one o Mutually exclusive Complement all outcomes in that are not in Denoted as Probability The probability for the event denotedis defined for any event and satisfies the following conditions 12 3 For any sequence of eventsthat are mutually exclusive then called countable additivity 4Note that o In general foreventsIfthen Ifnot necessarily disjoint then Probability space A probability space is the triple where we also useis the sample space of events subsets of is a set of subsets offorming a algebra on which we defineis a probability measure Example single coin flip We have the power set LetsoConditional probability The probability that occurs given that has occurred is denoted byOnly defined whenIndependent events and are independent if or ifTwo events that are not independent are said to be dependentBayes Formula Suppose that are mutually exclusive exhaustive events in ie a partition Let Then for we have that
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