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.
1
vote
3answers
26 views
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 ...
0
votes
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}...
2
votes
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 ...
0
votes
0answers
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.
...
0
votes
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 ($\alpha - \frac{1}{n})^2 = \alpha ^ 2 - \frac{2 \alpha}{n} + \frac{1}{n^2}$} > ...
4
votes
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 ...
1
vote
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 ...
1
vote
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 $\vert J \vert < \...
-2
votes
0answers
27 views
Proving not complete statistics in Beta Distribution [on hold]
Suppose X~Beta(θ,θ),(θ>0), and let {X1,X2,…,Xn} be an iid sample, T=Π(Xi∗(1−Xi) is a sufficient statistic for θ.
Show T*(x)=(${Π_i(X_i)} \\ {Π_i(1-X_i)}$) is not complete.
THoughts:
Since $x\_i$ ...
1
vote
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 ...
1
vote
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 ...
2
votes
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
votes
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 ...
1
vote
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 ...
2
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 $...
1
vote
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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 $...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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. ...
1
vote
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
...
0
votes
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 ...
1
vote
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 $x$ ...
-1
votes
3answers
31 views
$\sup(c+A) = c + \sup A$
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 $...
1
vote
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 ...
-2
votes
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 ...
1
vote
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:
$$P(\mu|\{x_i\})={\Large\Pi_i}\Big{[} \frac{a^v\Gamma(v+\frac{1}{2})}{\sqrt
{2\pi}\Gamma(v)} (\frac{1}...
0
votes
1answer
11 views
Distribution of negative of standard normal variate
Let $X$ be a standard normal variate. Consider another variate $Y$ such that
$$Y = \begin{cases}
-X & \text{for $-2 < X< 2$} \\
X & \text{otherwise}.
\end{cases}$$
I need to check ...
2
votes
3answers
49 views
Stumped by a pretty basic fraction division
I'm self-studying through Stroud & Booths's amazing "Engineering Mathematics", 7th Edition, and am still on the "Arithmetic" section. Even though I've gone through the whole chapter and a lot of ...
2
votes
2answers
38 views
Constructing a probability space
I am trying to understand the proof that
$$\mathbb Eg(X) = \int_\mathbb R g(x)d\mu_X(x),$$
where $X$ is a random variable on a probabiliby space $(\Omega, \mathcal F, \mathbb P)$. It starts with the ...
0
votes
1answer
26 views
Rescale part of a matrix to bound the maximum eigenvalue module
I am working with a matrix $A \in \mathbb{R}^{4 \times 4}$ with structure:
$A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} & a_{1,4} \\ a_{2,1} & a_{2,2} & a_{2,3} & a_{2,4} \...
1
vote
1answer
33 views
The average time before a person find their group
Imagine there are $N$ people throwing a party. For any two of them, the time before they meet each other and stick together thereafter is independent, and obeys an exponential distribution whose $\...
0
votes
0answers
40 views
Derivative of $\Gamma (1)=-\gamma$
How to compute $\Gamma'(1)=-\gamma$ where $\gamma$ is Euler's constant i-e the limit of the series $(1+\frac12 +\frac13 +\frac14 +...+\frac1n)-\ln(n)$ where n $\rightarrow \infty$
$\gamma= 0....
1
vote
0answers
15 views
Coverting denary numbers into octal, binary and hexadecimal form
I'm self studying (after not touching any match for a good 15 years) from Stroud & Booth's amazing "Engineering Mathematics". (This example is from page 55 of the 7th ediction, F.138 of the first ...
1
vote
1answer
22 views
Regression model + expected value, variance and autocorrelation of the error term
Consider this regression model
$$Y_t=X_t\beta+\epsilon_t, ~~~~~~~~~~\epsilon_t \sim WN(0, \sigma^2_{\epsilon})$$
with 3 different specifications of the error term:
$\epsilon_t=\alpha_1\epsilon_{t-1}+...
1
vote
1answer
35 views
Linear regression - find $b$, $s^2$, $R^2$
Suppose you want to fit the model $Y=\alpha+\beta X+\epsilon$ but you don't have the full data set $\left[\begin{matrix}y&X\\\end{matrix}\right]=C$. Instead you only have:
$$C'C=\begin{bmatrix}
...
-1
votes
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 ...
4
votes
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 ...
1
vote
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!
1
vote
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) \...
1
vote
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)}=...
1
vote
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 $u , v$...
0
votes
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 ...
