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Questions tagged [self-learning]

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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Dividing higher-order algebraic expressions

I've been self-studying from Stroud & Booth's amazing "Engineering Mathematics", and am currently on the "Algebra" section. I've been working with division of algebraic expressions, and the book ...
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0answers
14 views

Product of N univariate Gaussians with shared variance

[I know that there is a very similar question but in that case the Gaussians all have different variances.] I am trying to work out the product of N univariate Gaussian distributions $\prod_\limits{n}...
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3answers
92 views

How to self study topology?

I'm a first year undergraduate student and I'm a math major. Currently, I'm taking an intro to analysis class and a linear algebra class. However, I often feel constrained by what I do in class and ...
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21 views

Quadratic equation, absolute value of roots strictly superior to 1 conditions

Let's consider the equation: \begin{align} 1 - \phi_1 z - \phi_2 z^2 = 0 \end{align} We want to find the conditions on $\phi_1, \phi_2$ for the roots to have an absolute value strictly superior to 1. ...
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2answers
37 views

To show that square of the supremum of the set {$t \in \mathbb{R} | t^2 < 2 $} cannot be greater than $2$

Given the set T= {$t \in \mathbb{R} | t^2 < 2 $}. Take $SupT = \alpha$ and assume that $\alpha ^ 2 > 2$ Now ($\alpha - \frac{1}{n})^2 = \alpha ^ 2 - \frac{2 \alpha}{n} + \frac{1}{n^2}$} > ...
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1answer
51 views

To prove that $ \cap_{n=1}^{\infty} (0 , \frac{1}{n}) = \emptyset $

$$ \bigcap_{n=1}^{\infty} \left(0 , \frac{1}{n}\right) = \varnothing$$ Now assume that intersection contains $b$. So, $ b < \frac{1}{n} \forall n \in N$. Since $b > 0$, so we have by ...
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2answers
29 views

Using Archimedian property to prove that infimum of set is $0$

Given set A = [ $\frac{1}{n} | n \in \mathbb{N}$] It seems to me i have to prove two things : $\frac{1}{n} \geq 0 , \forall n \in \mathbb{N}$ 2, Assuming $b$ be another lower bound , and so $b \leq ...
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0answers
13 views

For gaussian process $X_t $ the best prediction is linear

I want to understand the proof for this Theorem: For gaussian process $X(t), t\in I $ the best prediction is linear. Proof: We only show the theorem for $ J \subset I $ with $\vert J \vert < \...
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0answers
27 views

Proving not complete statistics in Beta Distribution [on hold]

Suppose X~Beta(θ,θ),(θ>0), and let {X1,X2,…,Xn} be an iid sample, T=Π(Xi∗(1−Xi) is a sufficient statistic for θ. Show T*(x)=(${Π_i(X_i)} \\ {Π_i(1-X_i)}$) is not complete. THoughts: Since $x\_i$ ...
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2answers
60 views

About summation with p-adic valuation

Here is a problem i came across. Prove that $$\sum_{i=1}^{n}\frac{1}{i}$$ is not an integer for $n \geq 2$. The book olympiad number theory by Justin Stevens says that after writing $\frac{1}{i}$ as ...
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1answer
12 views

Continuous and Discontinuous Functions - Variation of Dirichlet function

Be $f: R \longrightarrow R $ a real function where: $ f(x) = \begin{cases} x + 1, & \text{if $x \in Q $} \\[2ex] 2, & \text{if $x \in R-Q$} \end{cases} $ Where the graph "jumps" among the ...
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3answers
29 views

How do I get from log F = log G + log m - log(1/M) - 2 log r to a solution withoug logs?

I've been self-studying from Stroud & Booth's excellent "Engineering Mathematics", and am currently on the "Algebra" section. I understand everything pretty well, except when it comes to the ...
2
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0answers
40 views

How can I calculate Feigenbaum's constant?

So I am trying to calculate Feigenbaum's constant for the logistic map: $$ x_{n+1} = 4 \lambda x_n (1-x_n) $$ I am writing this through python and the main pieces I have for my code that are relevant ...
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0answers
25 views

Showing the matrix is non-negative definite

Let $X$ be full column rank. I am trying to show that the matrix $$(X^TX)^{-1}(X\beta)^T(X\beta)-\beta\beta^T$$ is non-negative definite. Here, $\beta$ is a parameter vector with intercept. To ...
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0answers
40 views

How to reconcile difference in notation used in probability and statistics by different authors

After learning probability for so many years, I still have trouble with the notation whenever I encounter a new reference. I have pinpointed my confusion to this "two-culture" of probability: one is ...
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1answer
36 views

Introduction to Stochastic Integration book

I know this question has been asked several times (see: here, here, here, here, and here), but what are the "best" books on stochastic calculus: has anyone had experience with enough books on this ...
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0answers
41 views

In this constrained minimization problem, should the Lagrange multipliers be positive?

Consider the following (real, block ?) matrix $Z_{n\times k+1}=[1_{n\times 1},X_{n\times k}]$. Note how $z\equiv v^TZZ^Tv$ can be written as: $v^T11^Tv+v^TXX^Tv=v^TJ_nv+v^TWv$, where $J_n$ is a unit ...
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1answer
11 views

Proportionality term in Normal-Gamma distribution

I am currently learning from Christopher Bihops's Pattern Recognition and Machine Learning book about posterior distributions for the Normal distribution whenever both $\mu$ and $\tau$ (the precision ...
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0answers
12 views

Big O of Sum and Product

If I want to calculate the complexity of $X\times X$ the Cartesian product it is $O(n^2)$ with $n = |X|$. Let's say I can partition $X$ into $k>1$ equally sized subsets $Y_i$ with $m = |Y_1|$ and $...
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1answer
12 views

To prove special case when a <0

Original Theorem : Given two real numbers $a$ and $b$ with $a < b$, where $a \geq 0$ there exists a rational number $r$ satisfying $a < r < b$. To be proven from above theorem : Without ...
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1answer
40 views

Understanding proof of density of $\mathbb{Q}$ in $\mathbb{R}$

I am trouble getting this proof and idea behind it. I have taken example of a and b as (3,4). I have problem with understanding the part where author writes " first step is to choose n large enough so ...
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1answer
45 views

Obtaining a second linearly independent solution in an ODE using a known one.

Given the following problem: $\dot{x}(t) = A(t)x(t)$, say A is two by two. We are given a solution $\xi_1$. How can I use this in order to find a second linearly independent solution? Is there a ...
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2answers
63 views

Poisson Conditional Expectation ( searching best estimator for h(λ) )

Suppose $X_1$,$X_2$,$X_3$,.....,$X_n$ are i.i.d. random variables with a common density poisson(λ) (I is an indicator function) (t = a value) E $[$$X_2$ - I{$x_1$=1}|$\sum_{i=1}^n X_i=t$$]$ =E $...
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0answers
34 views

Finding out the sequence as Martingale.

Consider the sequence $\{X_n\}_{n\geq 1}$ of independent random variables with law $N(0, \sigma^2)$. Define the sequence $Y_n= \exp \bigg(a\sum_{i=1}^n X_i-n\sigma^2\bigg), n\geq 1,$ for $a$ a real ...
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1answer
13 views

Intuition behind MLE - Why is the MLE of inverse theta function the maximum X and not the minimum.

So I'm trying to understand why the MLE of $\theta$ for $$f_X(x) =\frac{1}{\theta}, \ \ 0\leq x\leq\theta$$ intuitively. For reference, I am using Example 5 from this paper that I found online. The ...
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3answers
19 views

Finding a unified ratio from two separate ratios

I'm self-studying with Stroud & Booth's amazing Engineering Mathemathics, 7th edition. I'm stuck at a problem set that gives me two ratios of variables A and B, and B and C respectively, an ...
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0answers
35 views

Which of following statements are correct regarding supremums and infimums?

A finite, non empty set always contains its supremum Correct as in a finite set no elements are same and some element is highest If $a < L , \forall a \in A$ then Sup A < L Incorrect. ...
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3answers
44 views

Prove that there exists $b \in B$ that is an upper bound for $A$

Given that $ \ Sup A < \ Sup B$ Now $\forall \epsilon, \exists b \in B \ such \ that $ $ \ Sup A - \epsilon < \ Sup B - \epsilon < b$ SO, $ \ Sup A - \epsilon < b $ So we have now ...
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1answer
20 views

argmin with logic statements [solved]

Could anyone explain me the equation below? Are we trying the find the T value where the $\theta_T < E_0[t]?$ $T^* = \underset{T}{Argmin} (\theta_T \geq E_0[T])$ I understand how argmin works for ...
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1answer
34 views

Prove that if $a$ is an upper bound for $A$, and if $a$ is also element of $A$, then $a = \ sup A$

Assume that $b$ is supremum of set $A$, where $b > a$. Let $x \in A$. So we have $x \leq a < b$. Since b is supremum for set A. So let us choose $\epsilon = b-a $ and so we must have some $x$ ...
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$\sup(c+A) = c + \sup A$

Let $A \subseteq \mathbb{R}$ be bounded above and let $c \in \mathbb {R}$. Define the set $c + A = \{c + a : a \in A\} $ Now since $a \leq \sup A , \forall a\in A$. Then $a + c \leq \sup A + c $. So $...
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2answers
80 views

To prove $\sup B = \inf A$

Let $A$ be bounded below and $B = \{b \in \mathbb{R} : b$ is a lower bound for $A\}$. My Work Now to understand I assumed some values, like set $A = (1,4)$ and $\inf A=1$ and so set $B = (-\infty ...
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2answers
57 views

Are there non-standard roads to research available for gifted students?

This is my first post here so I hope this is the right place to ask about this. I am a 22 year old math fanatic in Sweden and have been put in the awkward position of having self-studied quite ...
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0answers
20 views

smart way to calculate expected value of a difficult expectation

I have a non-normalized posterior distribution of which I'd like to calculate the expected value: $$P(\mu|\{x_i\})={\Large\Pi_i}\Big{[} \frac{a^v\Gamma(v+\frac{1}{2})}{\sqrt {2\pi}\Gamma(v)} (\frac{1}...
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1answer
11 views

Distribution of negative of standard normal variate

Let $X$ be a standard normal variate. Consider another variate $Y$ such that $$Y = \begin{cases} -X & \text{for $-2 < X< 2$} \\ X & \text{otherwise}. \end{cases}$$ I need to check ...
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3answers
49 views

Stumped by a pretty basic fraction division

I'm self-studying through Stroud & Booths's amazing "Engineering Mathematics", 7th Edition, and am still on the "Arithmetic" section. Even though I've gone through the whole chapter and a lot of ...
2
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2answers
38 views

Constructing a probability space

I am trying to understand the proof that $$\mathbb Eg(X) = \int_\mathbb R g(x)d\mu_X(x),$$ where $X$ is a random variable on a probabiliby space $(\Omega, \mathcal F, \mathbb P)$. It starts with the ...
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1answer
26 views

Rescale part of a matrix to bound the maximum eigenvalue module

I am working with a matrix $A \in \mathbb{R}^{4 \times 4}$ with structure: $A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} & a_{1,4} \\ a_{2,1} & a_{2,2} & a_{2,3} & a_{2,4} \...
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1answer
33 views

The average time before a person find their group

Imagine there are $N$ people throwing a party. For any two of them, the time before they meet each other and stick together thereafter is independent, and obeys an exponential distribution whose $\...
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40 views

Derivative of $\Gamma (1)=-\gamma$

How to compute $\Gamma'(1)=-\gamma$ where $\gamma$ is Euler's constant i-e the limit of the series $(1+\frac12 +\frac13 +\frac14 +...+\frac1n)-\ln(n)$ where n $\rightarrow \infty$ $\gamma= 0....
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0answers
15 views

Coverting denary numbers into octal, binary and hexadecimal form

I'm self studying (after not touching any match for a good 15 years) from Stroud & Booth's amazing "Engineering Mathematics". (This example is from page 55 of the 7th ediction, F.138 of the first ...
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1answer
22 views

Regression model + expected value, variance and autocorrelation of the error term

Consider this regression model $$Y_t=X_t\beta+\epsilon_t, ~~~~~~~~~~\epsilon_t \sim WN(0, \sigma^2_{\epsilon})$$ with 3 different specifications of the error term: $\epsilon_t=\alpha_1\epsilon_{t-1}+...
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1answer
35 views

Linear regression - find $b$, $s^2$, $R^2$

Suppose you want to fit the model $Y=\alpha+\beta X+\epsilon$ but you don't have the full data set $\left[\begin{matrix}y&X\\\end{matrix}\right]=C$. Instead you only have: $$C'C=\begin{bmatrix} ...
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1answer
29 views

Confidence interval for Poisson variables

Let $X_{i},...,X_{n}$ be i.i.d. Poisson random variables with parameter $\lambda>0$ I have: $$\bar{X}={(1/n)\sum_{i=1}^n X_i}.$$ Find two sequences $(a_n)_{n>=1}$ and $(b_n)_{n>=1}$ such ...
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0answers
177 views

Which books to use for self-studying calculus and linear algebra? [closed]

After years of divorce with mathematics I decided to come back to it. I studied computer science, so had quite a lot of maths in university, but though passing the exams, I didn't feel I fully grasped ...
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0answers
33 views

Graph theory and compact metric spaces: is it possible?

Is there a theory to study mathematical objects that are graphs but being its nodes higher dimensional compact metric spaces? Thanks!
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1answer
28 views

Quadratic form inequality?

I'm working on a problem, and to finish it I need to show that $$\mu^T (I-X(X^TX)^{-1}X^T)\mu \geq 0$$ But I'm stuck at showing this. My first thought was to write this as trace $$tr(\mu^T\mu) \...
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0answers
11 views

Separation of variables to solve Laplace's equation for the V in a cube 3D rectangular coordinates

Watch this videoLaplacian in 3D and tell me whether $C_{n,m}$ missing y in the argument of $\cosh{\sqrt{(\frac{n\pi}{a})^2+(\frac{m\pi}{a})^2}}$ in the denominator. So the final answer is $V_{(x,y,z)}=...
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1answer
14 views

Evaluating independence of vectors using Bilinear Forms

Let $f : U × U → \mathbb{R}$ be a bilinear form such that $f (u, u) > 0$ and $f (v, v) < 0$ for some $u, v \in U$. I would like to show that $u, v$ are linearly independent. We have that $u , v$...
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0answers
15 views

References books and lecture notes for Amenability

I am reading the book "Lectures on amenability" by Volker Runde I was wondering if someone could suggest me some books and Lecture notes with some good problems to go over My backgrounds are the ...