Foundations of Data Science

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1 − o(1) Probability of a giant component o(1) 1 − ε 1 + ε Expected number of friends per person Figure 4.1: Probability of a giant component as a function of the expected number of people each person knows directly. chain of connections, such as A knows B, B knows C, C knows D, ..., and Y knows Z, we say that A indirectly knows Z. Thus, all people belonging to a connected component of the graph indirectly know each other. Suppose each pair of people, independent of other pairs, tosses a coin that comes up heads with probability p = d/n. If it is heads, they know each other; if it comes up tails, they don’t. The value of d can be interpreted as the expected number of people a single person directly knows. The question arises as to how large are sets of people who indirectly know each other ? If the expected number of people each person knows is more than one, then a giant component of people, all of whom indirectly know each other, will be present consisting of a constant fraction of all the people. On the other hand, if in expectation, each person knows less than one person, the largest set of people who know each other indirectly is a vanishingly small fraction of the whole. Furthermore, the transition from the vanishing fraction to a constant fraction of the whole, happens abruptly between d slightly less than one to d slightly more than one. See Figure 4.1. Note that there is no global coordination of who knows whom. Each pair of individuals decides independently. Indeed, many large real-world graphs, with constant average degree, have a giant component. This is perhaps the most important global property of the G(n, p) model. 4.1.1 Degree Distribution One of the simplest quantities to observe in a real graph is the number of vertices of given degree, called the vertex degree distribution. It is also very simple to study these distributions in G (n, p) since the degree of each vertex is the sum of n − 1 independent random variables, which results in a binomial distribution. Since we will be dealing with 72
5 10 15 20 25 30 35 40 4 9 14 19 24 29 34 39 3 8 13 18 23 28 33 38 2 7 12 17 22 27 32 37 1 6 11 16 21 26 31 36 A graph with 40 vertices and 24 edges 17 18 19 7 9 20 21 22 23 6 8 24 25 3 4 5 10 26 27 1 28 29 2 30 31 11 12 13 14 15 16 32 33 34 35 36 37 38 39 40 A randomly generated G(n, p) graph with 40 vertices and 24 edges Figure 4.2: Two graphs, each with 40 vertices and 24 edges. The second graph was randomly generated using the G(n, p) model with p = 1.2/n. A graph similar to the top graph is almost surely not going to be randomly generated in the G(n, p) model, whereas a graph similar to the lower graph will almost surely occur. Note that the lower graph consists of a giant component along with a number of small components that are trees. 73
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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
Page 21 and 22: points onto the circle. In higher d
Page 23 and 24: Using the substitution 2z = y 2 , |
Page 25 and 26: For the nearest neighbor problem, i
Page 27 and 28: √ √ 2d+O(1) ≤ 2d + ∆2 −O(
Page 29 and 30: 2.10 Bibliographic Notes The word v
Page 31 and 32: Exercise 2.9 A 3-dimensional cube h
Page 33 and 34: Exercise 2.26 Explain how the volum
Page 35 and 36: Exercise 2.40 Consider a non orthog
Page 37 and 38: 1 Exercise 2.50 Use the probability
Page 39 and 40: This decomposition of A can be view
Page 41 and 42: 3.3 Singular Vectors We now define
Page 43 and 44: orthonormal basis w 1 , w 2 , . . .
Page 45 and 46: A n × d = U n × r D r × r V T r
Page 47 and 48: 3.6 Left Singular Vectors Theorem 3
Page 49 and 50: Lemma 3.10 (Analog of eigenvalues a
Page 51 and 52: Proof: Let A = r∑ σ i u i vi T i
Page 53 and 54: a i , let dist(a i , l) denote its
Page 55 and 56: such models. Mixture models are a v
Page 57 and 58: 1. The best fit 1-dimension subspac
Page 59 and 60: This leads to the following theorem
Page 61 and 62: Thus, the maximum cut problem can b
Page 63 and 64: and this meets the claimed error bo
Page 65 and 66: (0,3) (1,1) (3,0) ⎛ M = ⎝ 1 1 0
Page 67 and 68: Exercise 3.18 Modify the power meth
Page 69 and 70: 2. What percent of the Forbenius no
Page 71: City Bei- Tian- Shang- Chong- Hoh-
Page 75 and 76: Prob(vertex has degree k) = ( ) n
Page 77 and 78: and thus k k ≤ n. Since k! ≤ k
Page 79 and 80: For x to be equal to zero, it must
Page 81 and 82: No items E(x) ≥ 0.1 At least one
Page 83 and 84: Our first task is to figure out wha
Page 85 and 86: √ Thus, the probability for all s
Page 87 and 88: Figure 4.7: A degree three vertex w
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Page 91 and 92: Size of frontier 0 ln n nθ n Numbe
Page 93 and 94: For a small number i of steps, the
Page 95 and 96: same or are disjoint, ⎛( n∑ )
Page 97 and 98: are O(ln n) in size and the expecte
Page 99 and 100: f(x) q p 0 f(f(x)) f(x) x Figure 4.
Page 101 and 102: may be infinite. That is, ∞∑ i=
Page 103 and 104: Property cycles 1/n giant component
Page 105 and 106: Keep in mind that the leading terms
Page 107 and 108: 4.6 Phase Transitions for Increasin
Page 109 and 110: p(n) A symmetric argument shows tha
Page 111 and 112: simple. We supply some intuition be
Page 113 and 114: Proof: Say that t is a “busy time
Page 115 and 116: Consider a graph in which half of t
Page 117 and 118: problem contributes a minuscule err
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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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Exercise 5.10 Let p be a probabilit
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i 1 i 2 R 1 R 2 R 3 Figure 5.13: An
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(a) 1 2 3 4 (b) 1 2 3 4 (c) 1 2 3 4
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Exercise 5.40 Suppose that the cliq
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Exercise 5.55 Using a web browser b
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will allow us to make mathematical
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Not spam { }} { { Spam }} { x 1 x 2
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1 ∨ x 8 = 1}, or more succinctly,
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described using O(k log d) bits: lo
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In fact, we can show this bound is
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y definition of w ∗ . Similarly,
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y x φ Figure 6.4: Data that is not
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Theorem 6.11 (Online to Batch via R
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function of concept class H, will g
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Corollary 6.16 (VC-dimension sample
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A sphere in d-dimensions is a set o
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then with probability ≥ 1 − δ,
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work on a variety of measures. One
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Since weight(0) = n, the total weig
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Stochastic Gradient Descent: Given:
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sleeping experts problem. Combining
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⎫ ⎪⎬ ⎪⎭ Each gate is conn
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W 1 W 2 W 1 W 2 W 3 (a) (b) Figure
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convolution pooling Image Convoluti
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discard separators in advance that
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6.14.2 Active learning Active learn
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6.16 Exercises Exercise 6.1 (Sectio
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√ L log(L(T )) |S|) will be at mo
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7 Algorithms for Massive Data Probl
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important to know if some popular s
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We now give an example of a 2-unive
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estimate of m. To obtain a coin tha
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expected value of the sum will be z
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δ have values in ±1. There are ex
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mean. (ii) There is “no free lunc
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⎡ ⎢ ⎣ A m × n ⎤ ⎡ ⎥
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One uses the primitive in Section ?
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would require s ≥ n, which clearl
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its that capture sufficient informa
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7.6 Exercises Algorithms for Massiv
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Counting Frequent Elements The Majo
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Exercise 7.30 Blast: Given a long s
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Preliminaries: We will follow the s
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B A Figure 8.1: Example where the n
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Lloyd’s algorithm: Start with k c
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8.2.5 k-means clustering on the lin
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a clustering that is ɛ-close to C
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Theorem 8.5 Assume A satisfies (c,
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probability is q (where, q < p). 32
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8.6.4 Spectral Clustering Algorithm
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But for l ≠ l ′ , by hypothesis
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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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