Questions about the process of studying mathematics without formal instruction.

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1answer
15 views

Characteristic Function and Density Function

Consider a random variable $X$ with density function $f(x)$, moment generating function $M(t):= \int e^{tx}f(x) dx$ (existing in an interval containing $0$), cumulant generating function $K(t):=\log ...
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2answers
22 views

Find the distribution function of bivariate distribution

Find the distribution function of $$f_{X,Y}(x,y)=\begin{cases} e^{-y}, & \text{if $0< x<y < \infty$} \\ 0, & \text{ otherwise} \end{cases}$$ Trial : According to my calculation ...
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0answers
36 views

Seeking Recommendation on Number Theory textbooks [on hold]

S.E advisers, I am a college sophomore with double majors in mathematics and Russian language. I wrote this email to seek a recommendation on good introductory textbooks for number theory. I will ...
0
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1answer
49 views

How to solve this integral in moment generating function

The moment generating function of generalised Pareto distribution eventually comes down to the following integral (here). $$ M_X(\theta) = \mathbb Ee^{X\theta} = \int_\mu^\infty e^{\theta ...
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1answer
42 views

Matrix multiplication memorisation

So I'm writing an exam about matrices in a few weeks time, and I'd like to know if anybody has any tips about multiplying matrices.
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1answer
29 views

Requirements for learning and understanding trigonometry?

Im reaching a point in programing where I need to create basic shapes which I simply cant since my math skills are very bad. After finding out that the skills required are trigonometry I read a few ...
4
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2answers
65 views

Study a math course on my own, suggestions? [on hold]

I would like to study some math on my own. I am currently studying my second semester at a university and I have too much freetime so I would like to study something on my own, but I can't decide what ...
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0answers
20 views

Transformation Property of Bivectors

$\newcommand{\ba}[0]{\mathcal{B}}$ I want to derive the transformation property of a bivector ie. $$[\alpha]_{\ba'}=A[\alpha]_{\ba } A^T \tag{1}$$ where $[\alpha]_\ba$ denotes the matrix ...
1
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1answer
23 views

Concave and convex functions [on hold]

Does the function $ \frac{1}{x} $ in (0 ,10] , represent a convex function ? And the function $ log(x^{2}) $ , in the same interval, is concave function ? Thanks
1
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1answer
30 views

What textbook(s) do I need to self-study grade 9 - 11 math? I failed Gr 11 Math twice.

I need a book that will help me to master all the concepts in math up until at least toronto's grade 11 math level by June. It's not a long way away, and I'm a slow learner. So what can I do to speed ...
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0answers
18 views

Prove that minimum of the matrix norm is achieved at certain parametres

Given matrix $A\in R^{n\times m}$ prove that minimum of the $||A-xy^T||$, $||B||=tr(B^TB)$, is achieved when $x$ is an eigenvector of $AA^T$, corresponding to its greatest eigenvalue, and $y$ is an ...
1
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1answer
16 views

Inverse Laplace Transformation of a heaviside function.

I'm working through an example of an inverse laplace transformation: $$\mathscr{L}^{-1}[\frac{e^{-3s}}{s+1}] = u_3(t)e^{-(t-3)}$$ I am having trouble seeing how this works. I know that: ...
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0answers
23 views

Matrixproduct of A'A expressed as a sum

I have difficulties in proving (understanding, seeing) the following identity: $ \mathbf{A'A} = \mathbf{(a_1, a_2, ...,a_n)} \begin{pmatrix}\mathbf{a_1'\\a_2'\\ \vdots \\ a_n'}\end{pmatrix} = \sum ...
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0answers
26 views

Does Thompson's Calculus hit all the essentials needed?

I found out in my preliminary research for my undergraduate degree that I need a bit of undergraduate-level mathematics for my topic at the very minimum to start with, up to ODEs. So I found a ...
1
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1answer
35 views

Seeking Recommendation on Theoretical Multivariable Calculus textbooks

I am a college sophomore with double majors in mathematics and microbiology. I wrote this email to seek your advice on selecting a theoretical, proof-based textbook on the multivariable calculus. I ...
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1answer
70 views

Importance of Exercises in Mathematics for Self-Studying

I am a high school student wanting to major in Mathematics in the future. I started to like Mathematics recently, starting a year ago and I watched some interesting math videos on YouTube for fun (ex: ...
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1answer
17 views

Dual Vectors and Dual Metric

In the book of Nadir Jeevanjee „An Introduction to Tensors and Group Theory for Physicists“ it is stated as an exercise that: 2.17 Given a basis $\{e_i\}_{i=1,...,n}$ , under what circumstances do ...
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1answer
42 views

How to find conditional expectation $\mathbb E(X|X<M)$

Consider a random variable $X$ following the so-called folded normal distribution. That is, $X$ has density function $$ f_X(x) = \sqrt{\frac{2}{\pi\tau}}e^{-\frac{x^2}{2\tau}}, x>0. $$ ...
2
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1answer
71 views

The definition of the $false$ truth value

In "Topoi: The Categorial Analysis of Logic" by R. Goldblatt the $false: 1 \to \Omega$ truth value is defined as the characteristic arrow of the arrow $0_1: 0 \to 1$. This definition requires that ...
0
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1answer
43 views

What's in a name? (Sum of Squares)

I have always believed that in order to fully understand and appreciate the mathematical subtleties and ideas behind a certain concept is to understand the name given to it. For now, can someone ...
1
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1answer
18 views

Back Substitution

My professor explained the math behind a simple random walk process but I was unable to follow one of the steps. The first equation was: Xt=(Xt-1)+et He then said "we need to back substitute in for ...
5
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1answer
92 views

Complete example of haar measure on compact groups like $GL(n,R)$

I am currently reading the proof of existence of haar measure, but I learn better mostly by examples so I would like examples of explicit computation of haar measure mainly on any $Gl(n,R)$ or any lie ...
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4answers
124 views

Calculus books recommendation (intermediate level)

:) I would like to ask for some intermediate level textbook for calculus (single variable), or, at least, some supplement to Spivak's Calculus for better understanding on how to approach and solve ...
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0answers
25 views

Is this the right way to do it?

There are 40 questions. For each question, there are 5 options of which only one option is correct. 3 points are awarded for each correct answer, and 1 mark is deducted for each wrong answer. For ...
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1answer
35 views

On the good set principle and sigma fields.

Following Probability and measure Theory by Ash (2000). let $\Omega$ be a set, let $C$ be a class of subsets of $\Omega$ and $A \subset \Omega$, we denote by $C \cap A$ the class $\{ B \cap A : B \in ...
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0answers
19 views

Can anyone suggest a reference to learn about relative log-likelihood and likelihood intervals?

I want to understand how to calculate the 10% likelihood interval for a Poisson model of count data. It is an old assignment where they give you 20 counts, tell you it is a Poisson model and ask you ...
5
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2answers
63 views

How to minimize $x^2+4xy+5y^2-4x-6y+7$ without using calculus

I would like to find the smallest possible value of the function $$f(x,y)=x^2+4xy+5y^2-4x-6y+7$$ without taking any derivatives. My thoughts were to complete the square on both $x$ and $y$ and ...
0
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1answer
29 views

What is the number of words of length $h$ in a sequence of subsets of words?

Let $L=\{0,1\}^*$ (the set of binary words on $0$ and $1$), Given an integer $k$, and $S$ a finite subset of $L$ define recursively the following sequence of subsets of $L$: $$\begin{align} A_1 ...
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1answer
98 views

How to stay productive while you are studying math?

Not sure that this question is a good fit for this site, but I will try. When I am working through a chapter of a mathematical book first two hours are normally very productive (easily remember ...
6
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2answers
205 views

Prerequisits for Gauss-Green theorem

Consider the following theorem from the appendix C from Evans PDE book: I know about integration in $\mathbb{R}^n$ but not about how to make sense of the integrals on the right-hand side. As my ...
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0answers
12 views

Scheduling problem

Consider the following setting: $N$ jobs, each has a starting time, which is assumed to be a natural number and all N numbers are distinct, e.g., the 1st job has starting time at 5, the 2nd is 6, the ...
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0answers
10 views

Reference material on Alternating Minimization Algorithm

I am looking for some good reference material (book/paper) for learning Alternating Minimization Algorithm. Any recommendation from optimization experts will be much appreciated. Thank you.
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2answers
26 views

Question about an exercise from Feller

The following is an exercise from the classical textbook of Feller on probability theory. Four girls take turns at washing dishes. Out of the total of four breakages, three were caused by the ...
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4answers
70 views

Is $f\colon\mathbb{Z}\to\mathbb{Z}, f(x)=x^2$ injective? Surjective?

I would say no: $\text{Suppose } f(a)=f(b) \text{ then } a^2=b^2 \implies \pm a = \pm b \implies -a=b$. Or simply by counterexample: $f(-1)=f(1)$ Further, I would say it does not map $\mathbb{Z}$ ...
2
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1answer
44 views

Infinite horizon cost function

The following quote is from Bertsekas's Dynamic Programming and Optimal Control. I'm only looking for a nudge in the right direction as to how to interpret the following equations, particularly ...
6
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3answers
329 views

Equivalence Relation between Derivative Being Odd and Function Being Even

In the exercise, I am required to prove that $f'$ is odd $\iff$ $f$ is even Moving from right to left was pretty trivial, however, I couldn't move from left to right. Note that we can only use very ...
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2answers
20 views

If $A_n \downarrow A.$ then $A_1 - An \uparrow A_1 - A$? Set theory.

Let $A_1, A_2 , \dots$ be subsets of a set $\Omega$. If $ A_1 \subset A_2 \subset \dots$ and $\bigcup_{n = 1}^{\infty} A_n = A $ then we write $A_n \uparrow A.$ $ A_1 \supset A_2 \supset \dots$ and ...
0
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1answer
16 views

liminf and limsup in probability

Consider a sequence of random variable $(X_n)$. Prove the following inequality: $$ \mathbb P\left(\liminf\{X_n \leq x\}\right) \leq \liminf\mathbb P\left(\{X_n \leq x\}\right)\leq \limsup\mathbb ...
1
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1answer
31 views

Set Inclusion Properties

Consider a sequence of random variables $(X_n)$ converge almost surely to $X$. Define set $N:=\{\omega: X_n \to X \}^C$. Then it is claimed that we would have the following set inclusion properties. ...
2
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2answers
44 views

$4$ random digits, $2$ different ones.

I have been trying to solve the following problem What is the probability that among $4$ random digits, there appear exactly $2$ different ones? Two different digits means that there should be ...
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0answers
27 views

Find the distribution of $x_n$ ; $n\ge 1$, $x_1$ and $x_2$ also find the distribution $x_1, x_2,\dots , x_n$? [on hold]

Consider a general random walk, $$x_n = x_0 + y_1 +\dots + y_n$$ where the jumps $y_k$ are independent and $y_i \sim exp(\theta)$ Find the distribution of $x_n$ ; $n\ge 1$, $x_1$ and $x_2$ also find ...
2
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1answer
98 views

Proof that this specific function is measurable

Bounty Edit: Considering the nature of the problem at hand (i.e. proving that a specific function is measurable), I think this can be an easy but relevant problem. In particular, it is relevant to ...
2
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2answers
37 views

Prove cyclic group with one generator can have atmost 2 elements

Prove cyclic group with one generator can have atmost 2 elements . Attempt Consider a cyclic group generated by $a \neq e$ ie G = .So G is also generated by <$a^{-1}$> .Now Since it is given ...
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0answers
18 views

How to construct a two sided confidence interval?

A machine produces metal rods used in an automobile suspension system. A random sample of 15 rods is selected and the diameter is measured. The resulting data are shown below. 5.21 5.28 5.29 5.27 ...
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1answer
22 views

Consequence of linear combination in matrix .

If a column of a matrix is linear combination of another column, what are the consequences ? Several terminology coming into my mind to relate with this such as Rank of the matrix ; Determinant ...
1
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1answer
45 views

Showing that Determinant is a Volume Multiplier

I want to show using the change of change of variables theorem for (Riemann) integration that the determinant of a linear transformation $T$ is a scaling factor for the volume of a space. If $1_A$ is ...
1
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0answers
26 views

Convergence of Types Theorem

(Convergence of Types Theorem) Suppose that $F_n(u_nx+v_n) \Rightarrow F(x)$ and $F_n(a_nx+b_n) \Rightarrow G(x)$, where $u_n>0, a_n>0$ and $F$ an $G$ are non-degenerate. Then there exist ...
3
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1answer
61 views

$P( A\triangle B ) = 0 \Rightarrow P(A)=P(B)=P(A\cup B) + P(A \cap B) $

I want to prove the following statement; $$P( A\triangle B ) = 0 \Rightarrow P(A)=P(B)=P(A\cup B) \color{blue}{=} P(A \cap B) $$ What I did is that $$P(A\triangle B) ...
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3answers
51 views

$ P( A \mathrel{\triangle} B ) \le P(A \mathrel{\triangle} C) + P(B\mathrel{\triangle} C)$

Show that $$ P( A \mathrel{\triangle} B ) \le P(A \mathrel{\triangle} C) + P(B\mathrel{\triangle}C)$$ where $\mathrel{\triangle}$ indicates the symmetric difference I cannot write my idea, because ...
0
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1answer
27 views

Definition of Multiple .

Definition of multiple is : In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for ...