Foundations of Data Science

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6.15 Bibliographic Notes [TO BE FILLED IN] 232
6.16 Exercises Exercise 6.1 (Section 6.2 and 6.3) Consider the instance space X = {0, 1} d and let H be the class of 3-CNF formulas. That is, H is the set of concepts that can be described as a conjunction of clauses where each clause is an OR of up to 3 literals. (These are also called 3-SAT formulas). For example c ∗ might be (x 1 ∨¯x 2 ∨x 3 )(x 2 ∨x 4 )(¯x 1 ∨x 3 )(x 2 ∨x 3 ∨x 4 ). Assume we are in the PAC learning setting, so examples are drawn from some underlying distribution D and labeled by some 3-CNF formula c ∗ . 1. Give a number of samples m that would be sufficient to ensure that with probability ≥ 1 − δ, all 3-CNF formulas consistent with the sample have error at most ɛ with respect to D. 2. Give a polynomial-time algorithm for PAC-learning the class of 3-CNF formulas. Exercise 6.2 (Section 6.2) Consider the instance space X = R, and the class of functions H = {f a : f a (x) = 1 iff x ≥ a} for a ∈ R. That is, H is the set of all threshold functions on the line. Prove that for any distribution D, a sample S of size O( 1 log( 1)) ɛ δ is sufficient to ensure that with probability ≥ 1 − δ, any f a ′ such that err S (f a ′) = 0 has err D (f a ′) ≤ ɛ. Note that you can answer this question from first principles, without using the concept of VC-dimension. Exercise 6.3 (Perceptron; Section 6.5.3) Consider running the Perceptron algorithm in the online model on some sequence of examples S. Let S ′ be the same set of examples as S but presented in a different order. Does the Perceptron algorithm necessarily make the same number of mistakes on S as it does on S ′ ? If so, why? If not, show such an S and S ′ (consisting of the same set of examples in a different order) where the Perceptron algorithm makes a different number of mistakes on S ′ than it does on S. Exercise 6.4 (representation and linear separators) Show that any disjunction (see Section 6.3.1) over {0, 1} d can be represented as a linear separator. Show that moreover the margin of separation is Ω(1/ √ d). Exercise 6.5 (Linear separators; easy) Show that the parity function on d ≥ 2 Boolean variables cannot be represented by a linear threshold function. The parity function is 1 if and only if an odd number of inputs is 1. Exercise 6.6 (Perceptron; Section 6.5.3) We know the Perceptron algorithm makes at most 1/γ 2 mistakes on any sequence of examples that is separable by margin γ (we assume all examples are normalized to have length 1). However, it need not find a separator of large margin. If we also want to find a separator of large margin, a natural alternative is to update on any example x such that f ∗ (x)(w · x) < 1; this is called the margin perceptron algorithm. 1. Argue why margin perceptron is equivalent to running stochastic gradient descent on the class of linear predictors (f w (x) = w · x) using hinge loss as the loss function and using λ t = 1. 233
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Foundations of Data Science ∗ Avr
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4.4 Branching Processes . . . . . .
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8.2.2 Structural properties of the
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12.4.7 Median . . . . . . . . . . .
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densities, discrete optimization, e
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2 High-Dimensional Space 2.1 Introd
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Also, if x and y are independent, t
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1 1 − ɛ Annulus of width 1 d Fig
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integral gives ∫ ∞ 0 e −r2 r
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The volume of the hemisphere below
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points onto the circle. In higher d
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Using the substitution 2z = y 2 , |
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For the nearest neighbor problem, i
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√ √ 2d+O(1) ≤ 2d + ∆2 −O(
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2.10 Bibliographic Notes The word v
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Exercise 2.9 A 3-dimensional cube h
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Exercise 2.26 Explain how the volum
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Exercise 2.40 Consider a non orthog
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1 Exercise 2.50 Use the probability
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This decomposition of A can be view
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3.3 Singular Vectors We now define
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orthonormal basis w 1 , w 2 , . . .
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A n × d = U n × r D r × r V T r
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3.6 Left Singular Vectors Theorem 3
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Lemma 3.10 (Analog of eigenvalues a
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Proof: Let A = r∑ σ i u i vi T i
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a i , let dist(a i , l) denote its
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such models. Mixture models are a v
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1. The best fit 1-dimension subspac
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This leads to the following theorem
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Thus, the maximum cut problem can b
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and this meets the claimed error bo
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(0,3) (1,1) (3,0) ⎛ M = ⎝ 1 1 0
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Exercise 3.18 Modify the power meth
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2. What percent of the Forbenius no
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City Bei- Tian- Shang- Chong- Hoh-
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5 10 15 20 25 30 35 40 4 9 14 19 24
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Prob(vertex has degree k) = ( ) n
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and thus k k ≤ n. Since k! ≤ k
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For x to be equal to zero, it must
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No items E(x) ≥ 0.1 At least one
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Our first task is to figure out wha
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√ Thus, the probability for all s
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Figure 4.7: A degree three vertex w
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ftp://ftp.cs.rochester.edu/pub/u/jo
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Size of frontier 0 ln n nθ n Numbe
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For a small number i of steps, the
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same or are disjoint, ⎛( n∑ )
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are O(ln n) in size and the expecte
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f(x) q p 0 f(f(x)) f(x) x Figure 4.
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may be infinite. That is, ∞∑ i=
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Property cycles 1/n giant component
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Keep in mind that the leading terms
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4.6 Phase Transitions for Increasin
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p(n) A symmetric argument shows tha
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simple. We supply some intuition be
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Proof: Say that t is a “busy time
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Consider a graph in which half of t
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problem contributes a minuscule err
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one. On the other hand, for δ = 1,
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for all δ greater than δ critical
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expected degree d i = 1, 2, 3 √ t
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Destination Figure 4.17: For d < 2,
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For at least half of the pairs of v
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S i+1 is (1 − 1 ) |Si| ≤ 1 −
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Exercise 4.7 Let f (n) be a functio
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Exercise 4.19 (Birthday problem) Wh
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Exercise 4.38 Consider a model of a
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Exercise 4.53 Consider graph 3-colo
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5 Random Walks and Markov Chains A
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A B C (a) A B C (b) Figure 5.1: (a)
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in time polynomial in n. A quantity
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5.2 Markov Chain Monte Carlo The Ma
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p(a) = 1 2 p(b) = 1 4 p(c) = 1 8 p(
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5 8 7 12 1 3 1 6 1 8 1 3 3,1 3,2 3,
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Figure 5.4: A network with a constr
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There is a simple interpretation of
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Let γ i = π 1 + π 2 + · · · +
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5.4.1 Using Normalized Conductance
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The Gaussian distribution on the in
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6 6 1 8 1 5 8 4 5 5 Graph with boun
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and then reaching a from y before r
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every vertex? Hitting time The hitt
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clique of size n/2 x y } {{ } n/2 F
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i ↑ ↓ ↑ ↓ ↑ j ↑ =⇒ i
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Theorem 5.13 Let G be an undirected
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0 1 2 3 4 12 20 Number of resistors
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1 2 4 Figure 5.11: Paths obtained f
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whereas, with k self-loops, the equ
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or p[I − (1 − α)A] = α (1, 1,
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Page 187 and 188: Exercise 5.40 Suppose that the cliq
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Page 205 and 206: Theorem 6.11 (Online to Batch via R
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Page 215 and 216: work on a variety of measures. One
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Page 219 and 220: Stochastic Gradient Descent: Given:
Page 221 and 222: sleeping experts problem. Combining
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Page 225 and 226: W 1 W 2 W 1 W 2 W 3 (a) (b) Figure
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Page 237 and 238: 7 Algorithms for Massive Data Probl
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Page 241 and 242: We now give an example of a 2-unive
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Page 259 and 260: 7.6 Exercises Algorithms for Massiv
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Page 263 and 264: Exercise 7.30 Blast: Given a long s
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Page 267 and 268: B A Figure 8.1: Example where the n
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Page 279 and 280: 8.6.4 Spectral Clustering Algorithm
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4. the σ-interior of the clusters
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Figure 8.4: Example of a bipartite
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are likely to be balanced given tha
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s 1 ∞ ∞ λ t ∞ edges and vert
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A clustering algorithm is consisten
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less than n, then richness is not r
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8.13 Exercises Exercise 8.1 Constru
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A B Figure 8.8: insert caption Exer
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9 Topic Models, Hidden Markov Proce
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Nonnegative matrix factorization (N
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This is a linear program. As we rem
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for t = 1 to T Prob(O 0 O 1 · ·
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a ij transition probability from st
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C 1 S 2 C 2 causes D 1 D 2 diseases
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x 1 + x 2 + x 3 x 1 + x 2 x 1 + x 3
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x 1 x 1 + x 2 + x 3 x 3 + x 4 + x 5
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the messages coming to a variable n
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y 2 y 3 x 2 x 3 y 1 x 1 x 4 y 4 y n
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two pieces of information, the valu
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graph. The vertices of Π are denot
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present and ask how correlated this
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Now consider a very tall tree. If t
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9.14 Exercises Exercise 9.1 Find a
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10 Other Topics 10.1 Rankings Ranki
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b . a . a b b b b . . b b a a b b .
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In the discrete case, x = [x 0 , x
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Figure 10.3: Some subgradients for
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Suppose ˜x 0 were another minimum.
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A T S A S is invertible. We will pr
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was included, we would just multipl
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position on genome trees = Phenotyp
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form a matrix, called the Hessian,
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The simplex algorithm is a classica
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This problem is NP-hard. One way to
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10.9 Exercises Exercise 10.1 Select
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Exercise 10.18 Repeat the above exe
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Lemma 11.1 If a dilation equation i
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The Haar Wavelet ⎧ ⎨ 1 0 ≤ x
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11.3 Wavelet Systems So far we have
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11.5 Conditions on the Dilation Equ
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the support of both sides of the eq
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and the wavelet functions are ortho
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11.7 Sufficient Conditions for the
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Example of orthogonality when wavel
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11.9 Designing a Wavelet System In
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3. f(x) = 1 2 f(2x) + 1 2 f(2x −
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12 Appendix 12.1 Asymptotic Notatio
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these equalities by derivatives.
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Gaussian and related integrals To v
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1 + x ≤ e x for all real x (1 −
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Thus, n! ≤ n n e −n√ ne. For
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from which the current inequality f
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Example: Let f (x) = x k for k an e
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Random variables x 1 , x 2 , . . .
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12.4.7 Median One often calculates
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The density of y is the unit varian
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Generation of random numbers accord
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Example: Consider flipping a coin 1
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Setting λ = ln(1 + δ) Prob ( s >
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12.5 Bounds on Tail Probability Aft
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Collect terms of the summation with
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The first integral is just the stan
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of a symmetric matrix, counting mul
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capture this situation. Now AA T =
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The above theorem tells us that the
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Important special cases are |x| 0 t
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Lemma 12.23 Let A be a symmetric ma
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etween vertices in different blocks
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∞∑ ∑ ix i = ∞ i=0 i=0 x d d
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Generating functions are useful for
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Thus, u n = (n − 1) u n−2 . The
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f(x) a c b Figure 12.3: Illustratio
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need to argue that no other sequenc
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1. How large must δ be if we wish
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Exercise 12.40 We are given the pro
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Index 2-universal, 240 4-way indepe
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Lagrange, 423 Law of large numbers,
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References [AK] Sanjeev Arora and R
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[MR95b] [MU05] [MV10] Rajeev Motwan
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