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    Why does conditional probability used in mean square error equal zero?

    Hi guys, I am having trouble showing that \mathbb{E}\left[(Y-\mathbb{E}[Y|X])^{2}\right]=0. I understand the proof of why E[Y|X] minimizes the mean square error, but I cannot understand why it is then equal to zero. I tried multiplying out the square to get...
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    Dirac Delta/Mean Value Theorem Problem

    Homework Statement Consider the function \delta_{\epsilon}(x) defined by \delta_{\epsilon}(x)=\begin{cases} 0\text{,} & x<-\epsilon\text{,}\\ \frac{3}{4\epsilon^{3}}(\epsilon^{2}-x^{2})\text{,} & \epsilon\leq x\leq\epsilon\text{,}\\ 0\text{,} & \epsilon<x\text{,} \end{cases} (b) Consider a...
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    Transpose of the product of matrices problem

    Hi, The following equations are from linear regression model notes but there is an aspect of the matrix algebra I do not get. I have, \mathbf{y} and \tilde{\beta} are a mx1 vectors, and \mathbf{X} is a mxn matrix. I understand the equation...
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    Greatest lower bound problem - Rudin POMA Ch1 Exercise 5

    Homework Statement 5. Let ##A## be a nonempty set of real numbers which is bounded below. Let ##-A## be the set of all numbers ##-x##, where ##x\in A##.Prove that $$\inf A=-\sup(-A)\text{.}$$ Homework Equations The Attempt at a Solution Does the proof below look OK? I am a bit uneasy...
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    Limit help needed for end of complex number question

    I just realized I'm over complicating things. From the question and with ##k=nq## we have $$w_1^k+w_2^k+...+w_n^k=w_1^{nq}+w_2^{nq}+...+w_n^{nq}=(w_1^n)^q+(w_2^n)^q+...+(w_n^n)^q=(1)^q+(1)^q+...+(1)^q=n.\Box$$ In my original solution I have subtracted ##\pi## from the arguments of the...
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    Limit help needed for end of complex number question

    Hi tiny-tim, I can see that for Case 2 its easer to use $$S_n=e^{i\pi\frac{(2-n)k}{n}}\left(1+e^{i\pi\frac{2k}{n}}+...+e^{i\pi \frac{2(n-1)k}{n}}\right)$$ instead of reducing this to include the sin functions. Substituting in ##k=nq## gives...
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    Limit help needed for end of complex number question

    Homework Statement 43. Let ##w_1, w_2, ... , w_n## be the ##n## distinct ##n##'th roots of unity ##(n\geq0)##. Show that if ##k## is an integer then $$w_1^k+w_2^k+...+w_n^k$$ equals ##0## or ##n##. Find the values of ##k## for which the sum is ##n##. Hint:Write the roots in polar form and...
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    Help wanted to show given formula for argument of given complex number is valid.

    Thanks Curious3141 that makes sense. At least I got LaTeX practise ;).
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    Help wanted to show given formula for argument of given complex number is valid.

    Homework Statement 37. Let ##z=\frac{i(1+is)}{1-is}## where ##s\epsilon\mathbb{R}##. (a) Show that $$\text{Arg}(z)= \begin{cases} \quad\frac{\pi}{2}+2\arctan s & \qquad \text{for}\quad s\leq1,\\ -\frac{3\pi}{2}+2\arctan s & \qquad\text{for}\quad s>1. \end{cases}$$ Homework...
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    Rudin 1.21 Problem understanding proof of unique positive root to the

    Hi All, I feel I understand this now. There were two things I did not have right in my head. 1) To prove the assumption yn > x is false all you need to do is find one instance when it is false. Having t ≥ y - k provides one instance. I was not recognizing this. I was getting...
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    Rudin 1.21 Problem understanding proof of unique positive root to the

    Firstly, the set E is defined: "Let E be the set of all positive real numbers t such that tn<x." Later on the proof goes: "Assume yn>x. Put k=yn-x / nyn-1. Then 0 < k <y. If t ≥ y - k, we conclude that yn-tn ≤ yn-(y-k)n < knyn-1 = yn-x. Thus tn>x, and t is not a member of E. It follows...
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    Rudin 2.41, why does an unbounded infinite set have no limit points?

    I get it now. Back in 2.7 he defines a sequence and whenever you see x_n for (n=1, 2, 3,...) you should read this as a sequence. Thanks.
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    Rudin 2.41, why does an unbounded infinite set have no limit points?

    Please let me know if this should be in the Homework section. Part of Rudin 2.41 says that for a set E in ℝk... "If E is not bounded, then E contains points xn with |xn|>n (n = 1, 2, 3,...)." I can understand the argument this far. I do not get the next sentence "The set S consisting of these...
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