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...
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...
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...
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...