Union Bound, Geometric Variables, Coupon collectors problem and - PowerPoint PPT Presentation

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Union Bound, Geometric Variables, Coupon collector’s problem and Minimum Cut Maria-Eirini Pegia Study Group Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Context 1 Section 1: Union Bound 2 Section 2: Geometric variables 3 Section 3: Coupon collector’s problem 4 Section 4: Minimum Cut Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

P ✣ ❇ P P ❼ ➁ Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Union Bound Boole’s inequality known as the union bound Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

P ✣ ❇ P P ❼ ➁ Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Union Bound Boole’s inequality known as the union bound for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Union Bound Boole’s inequality known as the union bound for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events Formally for a countable set of events A 1 , A 2 , A 3 , ... , we have P ( ✣ i A i ) ❇ P i P ❼ A i ➁ Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut George Boole ( 1815 - 1864 ) - English mathematician, philosopher and logician - differential equations, algebraic logic - author of The Laws of Thought (1854) Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut George Boole ”No general method for the solution of questions in the theory of probabilities can be established which does not explicity recognize, not only the special numerical bases of the science, but also those universal laws of thought which are the basis of all reasoning, and which, whatever they may be as to their essence, are at least mathematical as to their form” Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

✂ Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Countable and uncountable sets Let E set. Finite Set If E ① ❣ or E ✂ ➌ 1 , 2 ,..., n ➑ then E is called finite Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Countable and uncountable sets Let E set. Finite Set If E ① ❣ or E ✂ ➌ 1 , 2 ,..., n ➑ then E is called finite Countably infinity If E ✂ N then E countably infinite Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Countable and uncountable sets Let E set. Finite Set If E ① ❣ or E ✂ ➌ 1 , 2 ,..., n ➑ then E is called finite Countably infinity If E ✂ N then E countably infinite Countable Set If E is either a finite set or a countably infinite set, then E is called countable set. Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Countable and uncountable sets Let E set. Finite Set If E ① ❣ or E ✂ ➌ 1 , 2 ,..., n ➑ then E is called finite Countably infinity If E ✂ N then E countably infinite Countable Set If E is either a finite set or a countably infinite set, then E is called countable set. Uncountable Set If E is neither a finite set nor a countably infinite set, then E is called uncountable set (uncountably infinite set). Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

❺ ❃ Ô ✟ ✏ ❃ Ð � ✏ Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Examples: countable set Examples: ❺ Z is a countable (infinity) set. f : Z Ð � N ➣ ➝ ➝ f ❼ n ➁ � ➛ 2 n if n ❈ 0 ➝ 2 ❼ ✏ n ➁ ✔ 1 ➝ ↕ if n ❅ 0 Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Examples: countable set Examples: ❺ Z is a countable (infinity) set. f : Z Ð � N ➣ ➝ ➝ f ❼ n ➁ � ➛ 2 n if n ❈ 0 ➝ 2 ❼ ✏ n ➁ ✔ 1 ➝ ↕ if n ❅ 0 ❺ N x N is a countable (infinity) set. Let k ❃ N . fundamental theorem of arithmetic Ô ✟ k = 2 m ✏ 1 (2n - 1), m,n ❃ N f : N x N Ð � N f(n,m) = 2 m ✏ 1 (2n - 1) Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

➌ ➑ � ✆ � ✆ � ✆ ➯ Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Example: uncountable set The set [0,1] is uncountable Proof: ❺ [0,1] is not finite. ❺ [0,1] is not countable infinity. Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Example: uncountable set The set [0,1] is uncountable Proof: ❺ [0,1] is not finite. ❺ [0,1] is not countable infinity. Assume [0,1] is countable infinity. Then [0,1] = ➌ x 1 , x 2 , x 3 ,... ➑ - Divide [0,1] into 3 equal length intervals. � 0 , 1 3 ✆ , � 1 3 ✆ , � 2 3 , 1 ✆ 3 , 2 - Let x 1 ➯ I 1 . Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

➯ Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Example: uncountable set - Divide I 1 into 3 equal length intervals. - Let x 2 ➯ I 2 ( I 2 ❵ I 1 ). - I 1 ❛ I 2 ❛ I 3 ❛ ... such that x m ➯ I m and Length( I n ) = 1 � n � ➟ 0. 3 n Ð Heine–Borel theorem Ô ✟ n � 1 I n = ➌ x ➑ ❵ [0,1] ✜ ✔➟ So x = x k for some k. Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Example: uncountable set - Divide I 1 into 3 equal length intervals. - Let x 2 ➯ I 2 ( I 2 ❵ I 1 ). - I 1 ❛ I 2 ❛ I 3 ❛ ... such that x m ➯ I m and Length( I n ) = 1 � n � ➟ 0. 3 n Ð Heine–Borel theorem Ô ✟ n � 1 I n = ➌ x ➑ ❵ [0,1] ✜ ✔➟ So x = x k for some k. Contradiction ( x k ➯ I k ). So [0,1] is uncountable. Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Georg Cantor ( 1845 - 1918 ) - German mathematician - introduced the term countable set - invented set theory - Discrete Mathematics - proved that R are more numerous than the N Ô ✟ ➜ ”infinity of infinities” Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Cantor paradox Input: line segment AB, length 1, which is washed by rain completely vertically. We place shelters in AB with the following procedure. Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Cantor paradox Assume that we select randomly a point of AB and let as symbolized it by C = ➌ a point of AB that is washed ➑ C ❵ [0,1] Cantor set C is uncountable, compact (closed + bounded in Euclidean space) and has length 0. C’ = [0,1] ❷ C has length 1 k � 0 ❼ 2 3 ➁ k = 1 1 3 ✔ 2 9 ✔ 4 27 ✔ ... � 1 3 P ➟ Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut Cantor paradox ❺ Set Theory Ð � C is huge (uncountable) _\_ _/_ " ( ) ) ❺ Probability Theory Ð � C is nothing Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and