F x x n 1 x 1

• [PDF File]The Discrete Fourier Transform

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  P1 n=1 x[n]e |!n = X(z)j z=ej! x[n] = 1 2ˇ Rˇ ˇ X(!)e|!n d! Uniform Time-Domain Sampling x[n] = xa(nTs) X(!) = 1 Ts P1 k=1 Xa !=(2ˇ) k Ts (sum of shifted scaled replicates of Xa()) Recovering xa(t) from x[n] for bandlimited xa(t), where Xa(F) = 0 for jFj Fs=2 Xa(F) = Ts rect F Fs X(2ˇFTs) (rectangular window to pick out center replicate ...

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• [PDF File]ECE 302: Lecture 5.1 Joint PDF and CDF

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  c Stanley Chan 2020. All Rights Reserved. What are joint distributions? Joint distributions are high-dimensional PDF (or PMF or CDF). f X(x) |{z } one variable

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• [PDF File]MATH 401 - NOTES Sequences of functions Pointwise and ...

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  Thus, {fn} converges pointwise to the function f(x) = 1 on R. Example 4. Consider the sequence {fn} of functions defined by fn(x) = n2xn for 0 ≤ x ≤ 1. Determine whether {f n} is pointwise convergent on [0,1]. Solution: First of all, we observe that fn(0) = 0 for every n in N. So the sequence {fn(0)} is constant and converges to zero. Now ...

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• [PDF File]1 Review of Probability - Columbia University

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  A simple computation (utilizing X = X 1 +···+X n and independence) yields E(X) = np Var(X) = np(1−p) M(s) = (pes +1−p)n. Keeping in the spirit of (1) we denote a binomial n, p r.v. by X ∼ bin(n,p). 3. geometric distribution with success probability p: The number of independent Bernoulli p trials required until the first success yields ...

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• [PDF File]Exponential Distribution

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  mean 1/λ, the pdf of P n i=1 X i is: f X1+X2+···+Xn (t) = λe −λt (λt) n−1 (n−1)!, gamma distribution with parameters n and λ. 3. If X1 and X2 are independent exponential RVs

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• [PDF File]Legendre Polynomials and Functions

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  f(x)P n(x) dx n =0,1,2,3... Since P n(x) is an even function of x when n is even, and an odd function when n is odd, it follows that if f(x) is an even function of x the coefficients A n will vanish when n is odd; whereas if f(x) is an odd function of x, the coefficients A n will vanish when n is even.

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• [PDF File]Jiwen He 1.1 Geometric Series and Variations

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  Define a function f on the interval (−1,1) f(x) = X∞ k=0 xk = 1+x+x2 +x3 +··· = 1 1−x for |x| < 1 As the Limit f can be viewed as the limit of a sequence of polynomials: f(x) = lim n→∞ p n(x), where p n(x) = 1+x+x2 +x3 +···+xn. Variations on the Geometric Series (I) Closed forms for many power series can be found by relating ...

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• [PDF File]Chapter 5

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  f(x) = 1 x. We have |fn(x)| < n for all x ∈ (0,1), so each fn is bounded on (0,1), but their pointwise limit f is not. Thus, pointwise convergence does not, in general, preserve boundedness. Example 5.3. Suppose that fn: [0,1] → R is defined by fn(x) = xn. If 0 ≤ x < 1, then xn → 0 as n → ∞, while if x = 1, then xn → 1 as n ...

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• [PDF File]Chap. 5: Joint Probability Distributions

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  2 Sec 5.1: Basics •First, develop for 2 RV (X and Y) •Two Main Cases I. Both RV are discrete II. Both RV are continuous I. (p. 185). Joint Probability Mass Function (pmf) of X and Y is defined for all pairs (x,y) by

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• [PDF File]MATH 461: Fourier Series and Boundary Value Problems ...

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  f(x) ˘a0 + X1 n=1 h an cos nˇx L + bn sin nˇx L i; i.e., we can associate with f this Fourier series, butnot f is equal tothis Fourier series. The Fourier coefficientsof f, on the other hand,are never in doubt. They are given by a0 = 1 2L Z L L f(x)dx an = 1 L Z L L f(x)cos nˇx L dx; n = 1;2;::: bn = 1 L Z L L

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• [PDF File]CS 350 Algorithms and Complexity - Computer Action Team

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  a. x(n)=x(n − 1) +5 for n>1,x(1) = 0 b. x(n)=3x(n − 1) for n>1,x(1) = 4 c. x(n)=x(n − 1)+ n for n>0,x(0) = 0 d. x(n)=x(n/2)+ n for n>1,x(1) = 1 (solve for n =2k) e. x(n)=x(n/3)+1 for n>1,x(1) = 1 (solve for n =3k) 2. Set up and solve a recurrence relation for the number of calls made by F (n), the recursive algorithm for computing n!. 3.

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• [PDF File]Example 2. f x) = x n where n = 1 2 3 - MIT OpenCourseWare

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  Δy n n ) n = ( x+Δ )n (−x = x+ n(Δ )(n−1 O(Δ 2 −x = nx n −1+O(Δx) Δx Δx Δx As it turns out, we can simplify the quotient by canceling a Δx in all of the terms in the numerator. When we divide a term that contains Δx2 by Δx, the Δx2 becomes Δx and so our O(Δx2) becomes O(Δx). When we take the limit as x approaches 0 we get:

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• [PDF File]Section 1.5. Taylor Series Expansions

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  1 1¡x = X1 n=0 xn; for jxj < 1: (1) We call the power series the power series representation (or expansion) for the function f (x)= 1 1¡x about x =0: It is very important to recognize that though the function f (x)=(1¡x)¡1 is de &ned for all x 6= 1; the representation holds only for jxj < 1: In general,

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• [PDF File]1 Definition and Properties of the Natural Log Function

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  • For r = 0, f(x) = x0 = 1. • For r = −n, f(x) = 1 x n, x 6= 0. ⇒ x−1 = 1 x. • For r = p q rational, f(x) = y, x > 0, where yq = xp. f(x) = x 1 n is the inverse function of g(x) = xn for x > 0. ⇒ g f(x) = x1 n n = x. • Properties (r and s rational) xr+s = xr ·xs, xr·s = xr s, d dx xr = rxr−1, Z xr dx = 1 r +1 xr+1 +C, r 6 ...

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• [PDF File]Assignment-6

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  Assignment-6 (Due 07/30) 1.Let sequences f n and g n converge uniformly on some set EˆR to fand grespectively (a)Construct an example such that f ng n does not converge uniformly on E. Solution: Take f n = g n = x+ 1=nand E= R. Clearly f n;g n!xuniformly on R.Now f ng n = (x+ 1=n)2, and we claim that this does not converge uniformly to x2.To see this, we let h n(x) be the sequence of the ...

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