Recent content by EdMel

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...

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...

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...

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...

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...

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...

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...

Thanks Curious3141 that makes sense. At least I got LaTeX practise ;).

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...

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...

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...

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.

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...