This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 { ,, } for quantities well 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 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 ) 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. 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 inclusionexclusion 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 : R (3) Remember that 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 ....
View
Full
Document
This note was uploaded on 01/04/2012 for the course ISYE 6414 taught by Professor Staff during the Fall '08 term at Georgia Institute of Technology.
 Fall '08
 Staff

Click to edit the document details