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# lecture1 - ISYE 6414 Lecture 1 Prerequisites Dr Kobi...

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ISYE 6414 Lecture 1 Prerequisites Dr. Kobi Abayomi August 23, 2010 Summation and Product We often use roman letters { X, Y, x, y... } for things we hope to measure or model; greek letters { α, β, θ } for quantities we’ll infer from directly measured quantities. Summation Notation We work with indexed vectors alot, like: X = ( X 1 , X 2 , ..., X n ) or x = ( x 1 , x 2 , ..., x n ) or any stuff = ( stuff 1 , stuff 2 , ..., stuff n ) n X i =1 stuff i = stuff 1 + stuff 2 + · · · + stuff n 1

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is translated as: Start with stuff 1 and add it to stuff 2 and keep on adding until stuff n . The stuff to do ( n times) can be as simple to do as taking a bunch of numbers { x 1 , x 2 , ...x 3 } and dividing it by the total number, i.e. n X i =1 x i n = 1 n n X i =1 x i = x 1 + x 2 + · · · + x n n or something more complicated, like taking each of those numbers, subtracting some other number, squaring the result, and dividing that by n - 1 n X i =1 ( x i - μ ) 2 n - 1 Product notation Just like summation, but the indexed items are separated by multiplication in place of addition n Y i =1 x i = x 1 · x 2 · · · x n Probability Here are some facts on probability. 0 P ( E ) 1 The probability of any event is between zero and one. For any event, A , P ( A ) + P ( A c ) = 1 . P ( A B ) = P ( A ) + P ( B ) - P ( A B ) . If E can be decomposed into disjoint 1 events E 1 , E 2 , E 3 then P ( S i =1 E i ) = i =1 P ( E i ). P ( A B ) = P ( A ) + P ( B ) - P ( A B ) . This is the inclusion-exclusion principle . The general version is... 1 Two events, or sets, are disjoint if there is no commonality between them, i.e. no intersection. For example: Let our experiment be to record the light at a traffic intersection. Then A = { the light is green } , B = { the light is red } are disjoint. 2
P ( S n i =1 E i ) = n i =1 P ( E i ) - i<j P ( E i E j ) - i<j<k P ( E i E j E k )+ · · · ( - 1) n +1 P ( n i =1 E i ). Conditional Probability The conditional probability of an event B given the occurrence of another event A is P ( B | A ) = P ( A B ) P ( A ) (1) Bayes’ Rule Bayes’ Rule is a generalization of our definition of conditional probability. Remember the definition of conditional probability: P ( A | B ) = P ( A B ) P ( B ) . Just apply our definition of joint probability in the general case to the numerator and the law of iterated probability to the denominator. And Bayes’ rule is, for two events: P ( A | B ) = P ( A B ) P ( B | A ) P ( A ) + P ( B | A c ) P ( A c ) (2) Random Variables A random variable associates a numerical value with each outcome of an experiment. A random variable is a function from the sample space to real numbers. In notation: X : Ω R (3) Remember that Ω is the sample space of an experiment. Remember that a function, say f : A → B associates an element of A with an element of B . Discrete Random variable We call a random variable discrete if it has a countable number of values. 3

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A list, table, or functional specification of the numerical values of X , a random variable, and the probabilities associated with it is the probability distribution of a random variable .
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lecture1 - ISYE 6414 Lecture 1 Prerequisites Dr Kobi...

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