recitation1 Probability Recap

recitation1 Probability Recap - Recitation 1 Probability...

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Recitation 1 Probability Recap
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What we will cover Basic probability Definitions and Axioms. Random Variables – PDF and CDF. Joint distributions. Some common distributions. Independence. Conditional distributions.
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Probability Real world - Full of uncertainty. . Eg. I have to reach home by 7:30 pm. Can I take the 7:15 pm 61 C at CMU and reach? How much time will the bus take after I get it (possible delays due to traffic, roads, etc) What if the bus arrives late? Probability – A mechanism for decision making in the presence of uncertainty
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Basic Concepts A sample space S is the set of all possible outcomes of a conceptual or physical, repeatable experiment. ( S can be finite or infinite.) E.g., S may be the set of all possible outcomes of a dice roll: An event A is any subset of S. Eg., A= Event that the dice roll is < 3.
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Probability A probability P(A) is a function that maps an event A onto the interval [0, 1]. P(A) is also called the probability measure or probability mass of A . Worlds in which A is true Worlds in which A is false P(A) is the area of the oval Sample space of all possible worlds. Call it E Its area is 1
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Kolmogorov Axioms All probabilities are between 0 and 1 0 ≤ P ( A ) 1 P ( E ) = 1 P (Φ)=0 The probability of a disjunction is given by P ( A U B ) = P ( A ) + P ( B ) − P ( A ∩ B ) A B A B ¬A ¬B A B ?
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Why use probability? There have been attempts to develop different methodologies for uncertainty: Fuzzy logic Qualitative reasoning (Qualitative physics) In 1931, de Finetti proved that : If you gamble using them you can’t be unfairly exploited by an opponent using some other system
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Random Variable A random variable is a function that associates a unique number with every outcome of an experiment. Discrete r.v.: The outcome of a dice-roll: D={1,2,3,4,5,6} Binary event and indicator variable: Seeing a “6" on a toss X =1, o/w X =0. This describes the true or false outcome a random event . Continuous r.v.: The outcome of observing the measured location of an aircraft ϖ ϖ S X( ϖ ) i X obs X
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Probability distributions For each value that r.v X can take, assign a number in [0,1]. Like the probability measure defined earlier. Suppose X takes values v 1 ,…v n . Then, P(X= v 1 )+…+P(X= v n )= 1. Intuitively, the probability of X taking value v i is the frequency of getting outcome represented by v i
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Bernoulli distribution: Ber( p ) Binomial distribution: Bin(n,p) Suppose a coin with head prob. p is tossed n times. What is the probability of getting
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recitation1 Probability Recap - Recitation 1 Probability...

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