1. Prove the following for the normalizing factor of the t-density - Hint: Use Stirling's formula...
7. The last exercise in the handout entitled Some Common Functions. Use Stirling's formula to prove that - (n+1)) 2 8. Exercise 1 from the induction handout. Prove that for all n 2 1: 2 Do this twice =
Use the spectral density formula for linear filters to compute the spectral density for y(t) = 0.51" w(t – r) where w(t) is white noise with variance o 13-00 0 (Hint: decompose 0.51" w(t – r) = {0.5" (t – r) + w(t) +0.5" w(t – r)) 1=- r=-00 r=1
2. The following is a well-known result known as Stirling's formula: lim For this problem, you may assume we already know this formula is true. Use it to calculate the limits of the four sequences below (c) lim 2n)vn (d) lim a lim (2n)! 7L (b) lim(2mm n00 n
2. The following is a well-known result known as Stirling's formula: lim For this problem, you may assume we already know this formula is true. Use it to calculate the limits...
4. Stirling's Formula is the claim that n! n-o0 >1. V2nn(n/e)" In this exercise, we will show how this can be obtained from the Central Limit Theorem Recall that Exponential () if fx(x)= Ae ^x, x > 0; the corresponding mgf is Mx (t) ,t<^ X = and GA)".,1 ва xa-le-px, x> 0; the corresponding mgf is Mx(t) = X~T(a,B) if fx(x)= T , t <B (a) Argue that, if Xi ~Exponential(1), i = 1,2,..., all independent, then for every...
Suppose X has a Poisson distribution with E(X) = 10. Use Stirling's Formula to approximate P(X = 10).
6. With the standard notations for single factor ANOVA that were introduced in Lecture 3, prove that E(MSThe atment) σ2 + Σ n㎡/(a 1). Hint: Use the short-cut formula of SS from Page 10 and the distributions of yi, from Page 9 of Lecture 3. Treat ment
6. With the standard notations for single factor ANOVA that were introduced in Lecture 3, prove that E(MSThe atment) σ2 + Σ n㎡/(a 1). Hint: Use the short-cut formula of SS from Page...
5. Let f be defined on [0,1] by the following formula: 1 t = 1/n (n + N) 2n 0, otherwise (a) Prove that f has an infinite number of discontinuities in (0, 1). (b) Prove that f is nonetheless integrable on (0,1). (Hint: remember your geometric series!
Your turn: Compute the volume of the volume of revolution bounded by the parametric curve (t, 1/t) for t E [1, 0o). Then use the formula we just found to find its surface area. Does this mean you have found an object that contains less paint (volume) than it takes to paint its exterior (area)?
Your turn: Compute the volume of the volume of revolution bounded by the parametric curve (t, 1/t) for t E [1, 0o). Then use the...
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We did not include a normalizing factor in (8.11), so Ilpk 112-2π and the Fourier coefficients of an integrable function f E L1 (T) are defined by 2π (8.12) -ikx 2nJ_π 8.2 For xe (0, π), let g(x) = x (a) Extend g to an even function on T and compute the periodic Fourier coeffi cients clg] according to (8.12). (Note that the case k = 0 needs to be treated separately.) Show that the periodic series reduces to...
(b) Use the identities above and the formula for the sum of a geometric series to prove that if n is an integer and je[1,2,..., n) then sin2 (2ntj/n) = n/2 t-1 so long as jメ1n/21, where Ir] is the greatest integer that is smaller than or equal to x. We were unable to transcribe this image
(b) Use the identities above and the formula for the sum of a geometric series to prove that if n is an integer...