F x 1 sqrt log

• [PDF File]Introduction to Algorithms - Northeastern University

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  Theorem 9.5 If f(x) = a nxn + a n 1xn 1 + + a 1x+ a 0, then f(x) = O(n) a) f(x) = 17x+ 11 b) f(x) = x2 + 1000 c) f(x) = xlogx d) f(x) = x4=2 e) f(x) = 2x f) f(x) = bxcdxe 9.2.4 Logs, Powers, Exponents We’ve seen f(n) = O(nd). If d>c>1, then nc = O(nc). nc is O nd; but nd is not O(nc). log b nis O(n) whenever b>1. Whenever b>1, c and d are ...

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• [PDF File]Lecture 3: Solving Equations Using Fixed Point Iterations

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  0 using xn+1:= g(xn) as described in the process above. If (xn) ∞ 0 converges to a limit r, and the function g is continuous at x = r, then the limit r is a root of f(x): f(r) = 0. Why is this true? Assume that (xn) ∞ 0 converges to some value r. Since g is continuous, the definition of continuity implies that lim n→∞ xn = r ⇒ lim n ...

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• [PDF File]Z f x dx = 1 be a continuous r.v. f x - University of California, Los ...

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  1 f X(y 1 n): where, f X() is the pdf of X which is given. Here are some more examples. Example 1 Suppose Xfollows the exponential distribution with = 1. If Y = p X nd the pdf of Y. Example 2 Let X ˘N(0;1). If Y = eX nd the pdf of Y. Note: Y it is said to have a log-normal distribution. Example 3 Let Xbe a continuous random variable with pdf f ...

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• [PDF File]Convex Functions - University College Dublin

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  The function f(x) = 10xis convex. 1.4.3 Logarithms For y > 0, the logarithm of y, written log 10 (y) is the value of xsuch that 10x= y. The number of decimal digits in yis 1 + blog 10 (y). The logarithm is not a convex function, but this function is: f(x) = ˆ 1 x 0 log 10 (x) x>0 1.4.4 Piecewise Linear Functions f(x) = 8

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• [PDF File]Convex Optimization — Boyd & Vandenberghe 3. Convex functions

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  Restriction of a convex function to a line f : Rn → R is convex if and only if the function g : R → R, g(t) = f(x+tv), domg = {t | x+tv ∈ domf} is convex (in t) for any x ∈ domf, v ∈ Rn can check convexity of f by checking convexity of functions of one variable

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• [PDF File]Math 220A HW 6 Solutions - University of California, San Diego

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  6. Let Gbe a region and suppose that f: G!C is analytic and a2G such that jf(a)j jf(z)jfor all z2G. Show that either f(a) = 0 or ais constant. Solution: If f(a) 6= 0, then 1 f is de ned and analytic on G. Observing that j1 f(z) jis bounded above by j1 f(a) j, we can use the maximum modulus principal to say that 1 f is constant.

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• [PDF File]Log normal distribution (from X α β α β α - William & Mary

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  1.0 α=1,β=1 α=1,β=2 α=5,β=0.5 x f(x) The cumulative distribution function on the support of X is F(x)=P(X ≤x)= 1 2 + 1 2 erf √ 2(ln(x)−α) 2β! x >0, where erf(x)= 2 √ π Z x 0 e−t2 dt. The survivor function on the support of X is S(x)=P(X ≥x)= 1 2 − 1 2 erf √ 2(ln(x)−α) 2β! x >0. The hazard function on the support of ...

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• [PDF File]Graph Transformations - University of Utah

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  1 cf(x) (a,b) 7!(a, 1 cb) shrink vertically by 1 c f(x) (a,b) 7!(a,b) flip over the x-axis Examples. • The graph of f(x)=x2 is a graph that we know how to draw. It’s drawn on page 59. We can use this graph that we know and the chart above to draw f(x)+2, f(x) 2, 2f(x), 1 2f(x), and f(x). Or to write the previous five functions

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• [PDF File]Chapter 1. Algorithm Analysis - Donald Bren School of Information and ...

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  C-1.12 Show that log b f(n) is Θ(logf(n)) if b>1 is a constant. C-1.13 Describe a method for finding both the minimum and maximum of n numbers using fewer than3n/2 comparisons. Hint: First construct a group of candidate minimums and a group of candidate maximums. C-1.14 An n-degreepolynomialp(x) is an equationof the form p(x)= #n i=0 a ix i,

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• [PDF File]Lecture 8: Branches of multi-valued functions - University of Washington

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  f 1(U) is an open subset ofE whenever U is open. A multi-valued function f on E ˆC assigns a set of complex values to each z 2E, i.e. f(z) is a set of complex numbers. Examples: logz = logjzj+i arg(z) with domain E = Cnf0g. The multiple values of logz differ by k2ˇi p

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• [PDF File]Big-O Examples - Wrean

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  2 x). |f(x)| = |(x+5)log 2 (3x2 +7)| = (x+5)log 2 (3x2 +7), for all x > −5 ≤ (x+5x)log 2 (3x2 +7x2), for all x > 1 ≤ 6xlog 2 (10x2), for all x > 1 ≤ 6xlog 2 (x3), for all x > 10 ≤ 18xlog 2 x, for all x > 10 We conclude that f(x) is O(xlog 2 x). Observe that C = 18 and k = 10 from the definition of big-O.

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• [PDF File]Square Roots via Newton’s Method - Massachusetts Institute of Technology

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  be equivalent to Newton’s method to find a root of f(x) = x2 a. Recall that Newton’s method finds an approximate root of f(x) = 0 from a guess x n by approximating f(x) as its tangent line f(x n)+f0(x n)(x x n),leadingtoanimprovedguessx n+1 fromtherootofthetangent: x n+1 = x n f(x n) f0(x n); andforf(x) = x2 ...

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• [PDF File]Propagation of Errors—Basic Rules - University of Washington

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  3. If z = f(x) for some function f(), then –z = jf0(x)j–x: We will justify rule 1 later. The justification is easy as soon as we decide on a mathematical definition of –x, etc. Rule 2 follows from rule 1 by taking logarithms: z = x£y logz = logx+logy –logz = p (–logx)2 +(–logy)2 –z z = sµ –x x ¶2 + µ –y y ¶2 10/5/01 2

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• [PDF File]Introduction I Asymptotics Introduction cse235@cse.unl.edu Introduction I

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  Limit Method Example 1 - Proof B Continued. I Using algebra, lim n !1 2n 3n 2 3 n I Now we use the following Theorem without proof: lim n !1 = 8 1 I Therefore we conclude that the quotient converges to zero thus, 2n 2 O (3n) Limit Method Example 2 Example Let f(n) = log 2 n, g(n) = log 3 n2.Determine a tight inclusion of

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• [PDF File]Maximum Likelihood Estimation - University of Washington

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  Maximum Likelihood Estimation Eric Zivot May 14, 2001 This version: November 15, 2009 1 Maximum Likelihood Estimation 1.1 The Likelihood Function Let X1,...,Xn be an iid sample with probability density function (pdf) f(xi;θ), where θis a (k× 1) vector of parameters that characterize f(xi;θ).For example, if Xi˜N(μ,σ2) then f(xi;θ)=(2πσ2)−1/2 exp(−1

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• [PDF File]Topic 15: Maximum Likelihood Estimation - University of Arizona

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  The principle of maximum likelihood is relatively straightforward. As before, we begin with a sample X = (X 1;:::;X n) of random variables chosen according to one of a family of probabilities P . In addition, f(xj ), x = (x 1;:::;x n) will be used to denote the density function for the data when is the true state of nature.

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• [PDF File]Math 133 Taylor Series - Michigan State University

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  If we write the nth derivative of f(x) as f(n)(x), this becomes: f(x) = X1 n=0 c n(x a)n with coe cients c n= f(n)(a) n!: warning: The coe cients are constants with no x, so c 1 = f0(a), not f0(x). Proof. By hypothesis f(x) is analytic, so f(x) = P 1 n=0 c n(x a)n for some c n; we will derive the desired formula for these coe cients. Since f(a ...

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• [PDF File]Contents

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  A(N 1)f(N) A(M)f(M) + MX 1 n=N A(n)(f(n+ 1) f(n)) # + A(M)f(x) A(N 1)f(y) = X y

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