The process of studying mathematics without formal instruction. _Don't_ use this tag just because you were self-studying when you came across the mathematical question you're asking; it is only for when the fact that you're self-studying is what your question is _about_.

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Marginalizing multivariate-normal distribution canonical form

Regarding the problem of margenalization of canonical forms of multivariate gaussian distribution it was mentioned in probabilistic graphical models text book that $$\int{C(X,Y;k,h,g)}dY$$ is ...
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2answers
40 views

Expectation Functional in Lebesgue and Riemann Terms – Looking for a clarification

Here there is a really central problem I am having self-studying probability theory, that concerns the relation between the definition of expectation in Lebesgue terms and in Riemann terms. I will ...
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1answer
35 views

Negative Mean Square Error

For simple random sampling, I have calculated somemean square errors for ratio-type estimators such as Isaki estimator, and Prasad Singh estimator. But, Mean Square Errors i obtained are negative. ...
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52 views

Is It Worth It Working Out Every Practice Problem In Math? (Without a calculator)

I'm bouncing back between trig, algebra, and calc books. I've noticed that most of the problems at some point seem to distill into very tedious arithmetic. It is nice to have the prowess of ...
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45 views

Suitable reference for learning symplectic geometry

I am interested in studying symplectic geometry by myself and I'm looking for a good text to use as a reference in the way. I am a bit lost because I've found a lot of notes and books on the subject ...
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3answers
772 views

Does the phrase “If you don't use it, you lose it” apply to mathematics? [closed]

I'm asking this because I ran into the following particular situation: I took some calc courses over 2013, where I learned, amongst other things, to integrate some pretty nasty functions, and this ...
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1answer
32 views

writing a piecewise regression model as a linear model

lets write the following piecewise regression model $$y= \alpha_0 + \alpha_1 x +\epsilon ;\ \ x\le x_0 $$ $$ y=\beta_0 +\beta_1 x + \epsilon \ \ x\gt x_0$$ according to the variable $x_0$ is ...
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38 views

Self learning complex systems modeling

I am currently a senior studying mathematics. My program is heavily focused on pure maths, however, my interests lie in the area of applied mathematics. Specifically, I am interested in the study of ...
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28 views

Between Oksendal and Karatzas books

I finished Oksendal "SDE , an introduction with applications" , did most of the exercises (not all with success, but I read the solutions for those); my next project would be to read the Karatzas ...
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20 views

Is every Complex Square Matrix similar to its transpose? [duplicate]

I am aware that every complex square matrix is similar to its transpose but I am having a hard time proving this. Should I try to use the previously asked question listed at $A matrix is similar to ...
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1answer
48 views

If $B(t)$ is Brownian motion then prove $W(t)$ defined as follows is also Brownian motion

Let $B(t)$ be standard Brownian motion on $[0.1]$ Define $W(t)$ as follows $W(t) = B(t) - \int_0^t \frac{B(1)-B(s)}{1-s} \, ds$ Prove $W(t)$ is also Brownian motion So I'm not sure how to deal ...
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1answer
94 views

Proof that $(t_1, \dots, t_r) \mapsto \sum^{r}_{i=1} | t_i - \alpha_i|^p$ is continuous - Problem with Inequality

Bounty Edit: I already edited the question after some important comments. The questions I have are highlighted below the supposed proof. Any feedback or answer is most welcome. Thus, I just found a ...
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1answer
120 views

Who is a mathematician? [closed]

My first question in Math SE. Basically the question itself, who is a mathematician? Is it someone who solves problems on his leisure time or as a part of a job or even as a hobby? who researches ...
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1answer
39 views

Two questions on finding the equation of a parabola word problem- Klein's Calculus: An Intuitive and Physical Approach

I am solving the following word problem "A high voltage cable is supported by two towers 2800 feet apart and 348 feet high. The cable hangs in approximately the shape of a parabola, and the lowest ...
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25 views

Find the set of undominated strategies in Cournot duopoly

Consider a version of the Cournot doupoly game, where firms 1 and 2 simultaneously and independently select quantities to produce in a market. The quantity selected by firm $i$ is denoted $q_i$ and ...
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1answer
53 views

Unique Positive Definite Square Root of a Positive Definite Matrix

If $A$ be an $n\times n$ positive definite matrix, then there exists a unique positive definite matrix $B$ such that $B^2=A$. My question is how to get this $B$. What is the name of the algorithm for ...
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80 views

What is the expected amount of time until the chain is in state 4?

Consider the continuous-time markov chain with state space {1,2,3,4} and infinitesimal generator ...
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1answer
64 views

Topics of analysis beyond in $\mathbb{R}$ and beyond of functions of single real variable?

Could someone list the topics of analysis beyond in $\mathbb{R}$ and beyond of functions of single real variable that a new math graduate student should be familiar with? Also could someone list the ...
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77 views

Where to go after Halmos' *Naive Set Theory*

I'm in the process of finishing Halmos' Naive Set Theory, and I found the subject fascinating, so I would like to carry on reading about Set Theory when I'm done. From what I've been able to gather ...
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1answer
44 views

Bayesian Inference and Disease Testing

I've been working my way through an introduction to Bayesian Inference in a Statistical Physics textbook (Tobochnik and Gould, 2010 - available online, excellent book). I've run across a problem that ...
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2answers
82 views

I'd like to teach myself Algebra 2 through Calculus BC, where could I start?

Im a junior in high school, since I was young I've had a profound interest in Math, and I'd always looked forward to high school mathematics. Little did I know, Special education had taken away what ...
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1answer
18 views

Linear Algebra Expression

I have found the rank of M, the basis for the null space and evaluated M$\begin{pmatrix} 1\\ -2\\ -3\\ -4\end{pmatrix}$. But, I am having some trouble answering the last part of the question. Could ...
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1answer
82 views

Sum of positive infinity and negative infinity

Consider the following function of $\tau$: $$ h(\tau) := C_1 \ln\left(1-\frac{a}{\tau}\right) - C_2 \ln\left(1-\frac{b}{\tau}\right), $$ where $a > b>0$ and ...
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1answer
44 views

Is this an acceptable way to find an eigenvalue?

I have a matrix M where $$ M = \begin{pmatrix} -2 & 2 & 2 \\ 2 & 1 & 2 \\ -3 & -6 & -7 \\ \end{pmatrix} $$ and it has an eigenvector of ...
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27 views

Extension of Scalars, Tensor Prodcuts Vs Cartesian Products

My goal is to in a sense, create an additional set of "scalars" from the field of complex numbers, and the ring of integers. That will also maintain the property of being bilinear under bilinear forms ...
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1answer
18 views

Getting the rational function with given characteristics

The curve C has an equation $$y = \frac{ax^2+bx+c}{x+d},$$ where $a$, $b$, $c$, and $d$ are constants. The curve cuts the $y$-axis at $(0,-2)$ and has asymptotes $x=2$ and $y = x + 1$. From a ...
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2answers
59 views

Product of two sums over the same interval

I have some terms of an expression as sums but I would like to simplify the solution to an easier and less complicated one. What I have is $$ X = \sum_{k=0}^\infty \left(\frac{z}{5}\right)^k ...
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1answer
15 views

Self study-Common expectation and variance for sum of independent random variables

I am doing a problem that reads Suppose $X_1, X_2..., X_n$ are independent random variables with common expectation $\mu$ and variance $\sigma^2$. Let $S_n$=$X_1+X_2+...+X_n$. Find the expectation ...
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1answer
51 views

Proving the limit of $\frac{n!}{10^{n}}$ using definitions

$\cdot \lim \limits_{n \to \infty} \frac{n!}{10^n} = \frac{10!}{10^{10}} * (\frac{n!}{10^n})$ for all $n \ge 11$ So we must find $N(M)$ such that $\lvert \frac{10!}{10^{10}} * (\frac{n!}{10^n}) ...
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2answers
34 views

Proving the nested interval theorem

Theorem: Let $\{I_n\}_{n \in \mathbb N}$ be a collection of closed intervals with the following properties: $I_n$ is closed $\forall \,n$, say $I_n = [a_n,b_n]$; $I_{n+1} \subseteq ...
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43 views

Some advice on self studying [closed]

I'm currently studying Mechanical Engineering and of course doing that I've run into proof based mathematics. I would love to do classes for it but I don't want to kill my GPA for a non major course. ...
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54 views

Bolzano-Weierstrass Theorem proof question

Since $[a_n,b_n] \subset [x,y] \forall n$, we know that $Q = \{a_n\}^\infty_{n=1}$is bounded Let $t= \sup Q$ (which will be the accumulation point) Let $P$ be any neighborhood of $t$, so that there ...
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35 views

Solving a question on trigonometric series

I have stated the sum of the sum of the series (by geometric series) which is $$S_n= \frac{z(1-z^n)}{1-z}$$ I am trying to prove the second part of the question. However, I am unable to reach to ...
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1answer
21 views

This is the first half of proving Bolzano-Weierstrass theorem

Just making sure I'm on the right track so far Every bounded infinite set of real numbers has at least one accumulation point Pf: Let S be a bounded set. Since S is bounded, there are real numbers ...
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177 views

Is it acceptable to use reduced row echelon to show basis?

I am asked to show that {a, b, c} forms a basis for $\Bbb R^3$. I'm just wondering if it is acceptable to use reduced row echelon to show it since it is not shown that way in the marking scheme? ...
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1answer
40 views

Proof of sets A and B involving set theory, showing: $(B^c - AB)^c = B$

Use set algebra rules to show why the complement of $(B^c - AB)^c = B$ => Let x be an object Assume $x\in (B^c -AB)^c $ or $x \notin (B^c - AB)$ So then $x \in B $ but $x\notin AB$, therefore $x ...
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1answer
45 views

proving if $a \le b$ then $\sup(A) \le \inf(B)$

Let $A$ and $B$ be bounded sets of real numbers such that $a\le b$ for all $a \in A$ and for all $b \in B$. Show that $\sup(A) \le \inf(B)$ Pf: Assume A and B are bounded sets. This means they have a ...
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1answer
19 views

Proof using triangle inequality

Fix a real number x and a positive number $\epsilon$. If $\lvert x-1\rvert \le \epsilon$, show that $\lvert 2-x\rvert \ge 1 - \epsilon$ Pf: Fix $x \in R$ and let $\epsilon>0$, We know ...
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38 views

Proving that $sup(-A) = -inf(A)$

Let A be a bounded set of real numbers. Define $-A =$ {$x: -x \in A$}. Show that -A is bounded and that $sup(-A) = -inf(A)$. Pf: A is bounded so $\exists x,y \in A $ such that $inf(A) \le y <x ...
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2answers
43 views

Proving the limit of a sequence using definitions

This is a review that my professor posted and I want to make sure I'm on the right path as I study $\cdot \lim \limits_{n \to \infty} n - \sqrt{2n^2+1} = n ...
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3answers
124 views

mathematical rigore for an engineer! [closed]

I recently bought a used copy of "Mathematical Analysis" by Apostol for \$1.0 and "Probability and Measure Theory" by Robert Ash for \$3.0 (well another \$3.99 for shipping)! When I read the first few ...
3
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1answer
60 views

$\lim\limits_{A \to \infty} (1 - \frac{c}{A})^{N-1}$, $N \sim \mathrm{Poisson}(\lambda A)$

Given $N \sim \mathrm{Poisson}(\lambda A)$, what will be $(1 - \frac{c}{A})^{N-1}$, where $c$ is a constant, when $A \to \infty$? Will it converge into a distribution or a constant? And what is it? ...
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85 views

What are the minimum and maximum prerequisites to study Stochastic Processes?

Suppose, I never studied random variables. This is the syllabus: Lecture contents Review of important notions of probability theory (4h). A few remarks on stochastic processes : ...
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200 views

How can we prove $e^{\pi}-\pi\simeq 20$ geometrically? [closed]

Using a calculator we can easily check that $$\color{Green}{e^{\pi}-\pi}=19.999\cdots\color{Green}{\approx 20}$$ This article and this one provides some details about this almost near identity, but no ...
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1answer
63 views

Expected mean squared error and MSR

In a small-scale regression study, five observations on $Y$ were obtained corresponding to $X = 1,4,10, 11$, and $14$. Assume that $\sigma=0.6,B_0=5,B_1=3$ a. What are the expected values ...
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45 views

Undergrad looking for study material/advice for applied mathematics.

I am an undergraduate math student (junior) who is looking to get a masters degree in Applied Math. I like pure math, but I want to use my education to get a great-paying job. Here are a few questions ...
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53 views

Prove that OLS estimator of the intercept has minimum variance

Let $$y_i=B_0+B_1X_i+\epsilon_i$$ where $\epsilon_i\sim > N(0,\sigma^2)$. Find the least squares estimator of $B_0$ and show that it is unbiased and has minimum variance. I will not write in ...
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1answer
71 views

New to Abstract Algebra, need guidance [closed]

I am new to Abstract Algebra. How should I begin learning it. On the face if it, it doesn't look easy to me.I have bought the book by Prof. Gallian. Is there any other book or any videos which I can ...
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2answers
52 views

Convergence of fixed point iteration when $g'(p)=1$.

I am dealing with a function $f(x)=e^{-\frac{1}{x^2}}$, which has a root $p$ of infinite multiplicity at 0. I am struggling with the convergence rate of the resulting standard Newton fixed point ...
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1answer
37 views

Book recommendations for learning financial/business mathematics.

Does anyone know a book which covers topics on: Simple interest Compound interest Equations of equivalent values Nominal rate, effective rate and equivalent rate Annuities ...