Questions about studying mathematics without formal instruction.

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Question about Lebesgue Covering Dimension

Suppose we have a metric space equipped with two different metrics: $(X,d), (X, d')$. What must be true of the metrics: $d, d'$ in order for $X$ to have the same Lebesgue covering dimension? A ...
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1answer
37 views

Proof on existence of the natural numbers, crucial step.

I am trying to understand/reconstruct the proof given by my Professor addressing the existence of natural numbers. However there is one step in particular I don't understand and the more I think about ...
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1answer
29 views

Without solving the equation, determine the nature of its roots: $x^2 + ax + a^2 = 0$

I'm working through the book Core Maths for Advanced Level on my own, and, after solving the above problem, I'm not getting the same answer as the book. So, given: $$x^2 + ax + a^2 = 0$$ Using the ...
1
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1answer
39 views

Want to check measure theory proof

I need to show that sigma-finiteness implies semifinite. Does the following proof work? Let $(X,m,\mu)$ be a $\sigma-$finite measure. Let, $E\in m$, and $\mu(E)=\infty$. By $\sigma-$finiteness, ...
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3answers
456 views

Probably of 2 six in 5 dice rolls

What is the probability of obtaining exatcly 2 six when rolling a dice 5 times? In order to obtain this probability, I will need to devide the number of favorable events by the number of possible ...
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24 views

How to prove this claim in real analysis

E is a set of positive outer measure, how to prove that there must be a bounded subset of E that has positive measure?
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1answer
45 views

An exercise on first order logic formulas, terms and Polish notation

This is part of my homework (not mandatory and not accredited). Please comment/answer if my reasoning for the exercises is correct, because I'd like to see if I understand the material. I will start ...
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0answers
50 views

Help in Understanding the Formula for The Lattice Point Counting in Triangles with Rational Coordinates

Yesterday I have found this paper while searching Google. However, since the author of this paper gave no examples of implementing the following formula, I don't understand how to implement it in ...
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2answers
21 views

problem with rudin theorem 1.14 first step $g^{-1}((\alpha, \infty]=\cup_{n=1}^\infty f_n^{-1}((\alpha, \infty])$

Theorem 1.14 of Complex and real analysis by Rudin states: if $f_n:X\to [-\infty, +\infty]$ is measurable for $n = 1,2,3...$ and: $$g=\sup_{n\geq 1} f_n,~~~~h=\limsup_{n\to \infty} f_n,$$ then $h$ ...
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1answer
48 views

Show expectation is infinite

Let $X_1,\ldots,X_n$ be independent, identically distributed with expectation 1 and finite variance. Find the limit distribution of $\sqrt{n}(\bar{X}_n^{-1}-1)$. If the random variables are sampled ...
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1answer
87 views

The Use of Sound in Mathematics. [closed]

I'm not sure that this question is appropriate here. There's a good chance it's too opinion-based. If that's the case, I'm sorry. I was sat in a research seminar recently and wondered whether it'd be ...
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1answer
91 views

Proving, that closure of set is equal this set iff set is closed

I've started intorduction to topology course and I need help with one of the problems: Let $A \subset(X,T). $ Prove that $cl(A) = A\iff A$ is closed. It may looks trivial, but I had a little ...
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1answer
31 views

Definition of $C^1$ functions with values in $\mathbb R^m$

My analysis textbook defines a $C^1$ function $f:\mathbb{R^n}\to\mathbb{R^m}$ as one in which for each component function $f_i, 1\leq i\leq m$ the partial derivative $\frac{\partial f_i}{x_j}$ exists ...
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2answers
60 views

Path to 3d Mathematics programming, where to start?

This might read like duplicate of this question https://math.stackexchange.com/search?q=where+to+start However since that one wasn't answered, and I have a more specific problem in regards to ...
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1answer
26 views

Fourier Transforms of $L^1$ functions

Suppose that $f_n$ and $f$ are $L^1(\mathbb R^n)$ functions with $f_n \to f$ in $L^1$ sense. Then is it true that their Fourier transforms defined as $$ \hat f(\xi) := \int_{\mathbb R^n} ...
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0answers
43 views

How to Read Michael Artin's Algebra

I am currently reading Algebra by Michael Artin (for self study). I'm finding the book pretty interesting, but I'm still not sure how I should study the book. For example, should I try to prove every ...
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1answer
27 views

Conditional probability with balls in urns involving discards

I found this problem in a statistics book, and I'm wondering if my solution is correct. "You and a friend play a game involving 20 balls in an urn, of which 1 is red and 19 are white. The game is ...
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76 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
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1answer
18 views

Expected value of Cumulative Hazard

Define $T=\min(T^0,C)$ where $T^0$ is the failure time and $C$ is the censoring time. Define the failure indicator $$\delta = \begin{cases} 1 & \text{if $T^0\leq C$}\\ 0 & \text{if $T^0> ...
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1answer
33 views

Convergent Series $\frac{1}{n^q}, \ \ q>1$

How to show the following result about series? Thank you! Convergent Series: $$\sum_n \frac{1}{n^q}, \ \ q>1$$
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1answer
24 views

Finding probability given mean and standard deviation

I don't know how to approach this problem: X is normally distributed with a mean of 200 and a standard deviation of 10. Find P(X ≥ 203)
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3answers
40 views

Finding a recursive formula for a number

I am trying to find a recursive formula for a given number in order to solve a problem I am working on. For every $n \in \mathbb{N} \setminus \lbrace 0,1 \rbrace$ we define the number ...
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2answers
46 views

Mathematical background for one wishing to study Chaos/Complexity Theory

I don't have a very strong mathematics background. In fact I quite abhorred mathematics during my Middle/High School years. I'm currently applying for PhD programs in the field of literature as that ...
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0answers
25 views

On a step in a solution of a proof of the Continuity of thomae's function at irrationals.

Thomae's function: $$t(x) = \begin{cases} 0 \ \ \text{if} \ x \in \mathbb{R}- \mathbb{Q} \newline \frac{1}{q} \ \ \text{if} \ x \in \mathbb{Q} \ \text{and} \ x = \frac{p}{q} \ \text{in lowest terms} ...
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2answers
59 views

Good book for self-studying Binary Relations

I am studying economics and I frequently encounter Binary Relations. But without any good knowledge of it, I get confused. Here is some background, if it's helpful: I know calculus(single and ...
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2answers
34 views

Proof of: Because $I$ is dense in $R$ there exists a sequence $\{x_n\} \subset I$ with $\{x_n\} \to r$.

In a passage from some school slides I have written: consider an arbitrary $r \in Q$. Because $I$ is dense in $R$ there exists a sequence $\{x_n\} \subset I$ with $\{x_n\} \to r$. This seems ...
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0answers
26 views

Complex trigonometric equation

Find a solution to the equation $\tan(z)=7i$ which satisfies the condition $0<\Re(z)< \pi$} We use the $\sin(z)=\frac{e^{zi}-e^{-zi}}{2i}$ and $\cos(z)=\frac{e^{xi}+e^{-xi}}{2}$. Here is ...
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0answers
34 views

Joint density of $X_1^2+X_2^2\ \text{and} X_2,\ X_i\sim N(0,1)$

Let $X_1 $ and $X_2$ be iid with a common standard normal distribution. I am looking to find the joint pdf of $Y_1 =X_1^2 +X_2^2$ and $Y_2=X_2$. I know i can use a straight transformation argument but ...
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2answers
42 views

Product metric spaces is again a metric space

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let: $$ d_2 ((x_1,y_1),(x_2,y_2)) = \left[d_X(x_1,x_2)^2 + d_Y (y_1,y_2)^2 \right]^{\frac{1}{2}} $$ for the points $(x_1,y_1)$ and $(x_2,y_2)$ in $X ...
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2answers
118 views

Bartle vs Rudin, which one is better for real analysis?

I'm in high school and I want to study real analysis, and I can choose between The elements of real analysis by Robert G. Bartle and Principles of mathematical analysis by Walter Rudin, so, from the ...
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2answers
134 views

on a recursive sequence (exercise 8.14 Apostol).

The exercise asks to prove convergence and find the limit of the sequence:$$a_n= \frac{b_{n+1}}{b_n},\text{ where } b_1=b_2 =1, b_{n+2} = b_{n} + b_{n+1}. $$ It also gives a hint: Show that $ \ ...
2
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1answer
34 views

Possible number of arrangement.

Question: How many cars are there with number GJ-X-AB-abcd. GJ and A are constant.X is digit between 1 to 9, B is english alphabet and abcd is 4 digit number.(a can be zero) My Efforts: It is but ...
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1answer
17 views

Formally proving $\sum_{k=1}^{\infty}P\left(-k<X\leq-k+1\right)=P\left(X\leq0\right)$?

$\sum_{k=1}^{\infty}P\left(-k<X\leq-k+1\right)=P\left(X\leq0\right)$ This fact seems pretty obvious but how would I formally prove it, is there a painless way?
2
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1answer
26 views

Calculating a harmonic conjugate

Is the following reasoning correct? Determine a harmonic conjugate to the function \begin{equation} f(x,y)=2y^{3}-6x^{2}y+4x^{2}-7xy-4y^{2}+3x+4y-4 \end{equation} We first of all check $f(x,y)$ ...
0
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1answer
21 views

Case Deletion Diagnostics

I have NO idea how to approach this problem. I don't see any connection between the corollary and the formula we need to prove. Does anyone have any hints? Corrolary: If $\mathbf{A}$ and ...
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2answers
59 views

Proving formally $\lim_{x \to -\infty}\mathrm{Pr}( \left \lfloor{x}\right \rfloor \le X < x) = 0$ (Proof check)

we have $$\lim_{x \to -\infty}\mathrm{Pr}( \left \lfloor{x}\right \rfloor \le X < x) $$ where X is a real random variable, and $x \in R$. My idea of a proof would be by contradiction: Assume ...
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1answer
48 views

Examining the effect of a quantitative factor on response.

To examine the effect of a quantitative factor temperature on yield,the researcher has a plan to use the following model for the analysis: $$y_{ix}=\beta_0+\beta_1 x+\epsilon_{ix}$$ where $y_{ix}$ ...
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2answers
81 views

Parametric solution of the Diophantine equation $\frac{1}{p}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z} ,x,y,z∈Z^+.$

I have prove that, for any given positive integer $p,$ parametric solution of the Diophantine equation $$\frac{1}{p}=\frac{1}{x}+\frac{1}{y}$$ can be written in the form $x=ac(a+b),y=bc(a+b),$ where ...
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1answer
36 views

Why $\{Z \le z\} = \bigcap_{m = 1}^\infty \bigcup_{n=1}^\infty \bigcap_{k=n}^\infty \{ Z_k \le z + 1/m \}$ if $Z=\lim_nZ_n$?

I am following A first look at rigorous probability theory by Rosenthal, and I am having troubles with limits of random variables. Specifically proposition 3.1.5. (iii) states that if $Z_1,Z_2...$ ...
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40 views

What is the most “powerfull” method to prove a sequence is increasing or decreasing?

Given a sequence $a_n$ defined in a recursive manner, the methods I know to prove if the sequence is increasing are: 1) observe if $a_{n+1} - a_n > 0 \ \forall n.$ 2) take $\frac{a_{n+1}}{a_n}$ ...
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1answer
13 views

Testing statistic $\frac{MSS(X)}{MSS(Y)}$

Suppose a test statistic $\frac{MSS(X)}{MSS(Y)}$, where $MSS$ denotes Mean Sum of Squares, is to be used for testing the significance of the factor $X$. Do we need the assumption $$\mathbb ...
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1answer
68 views

general topology (self learning)

Hi everyone I'd like to know if the following is correct. I'd appreciate any suggestion. Thanks in advance. From Dudley´s book: Let $A_n$ be the set of all the integers greater than $n$. Let ...
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2answers
34 views

Determining if the relation is an equivalence one.

Determine if the relation : $$x \sim y \iff |y-x| \text{ is an integer multiple of } 3$$ is an equivalence one. Now, I think this is an equivalence relation but I am having troubles formally ...
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2answers
53 views

on the exercise 8.10 Apostol. (limit of sequence)

The exercise states: prove that the limit of the sequence $$a_{n+2}=(a_na_{n+1})^{1/2} \ where \ a_1 \ge 0, a_2 \ge 0 $$ is $L = (a_1a_2^2)^{1/3}$ The solutin says: $$Let \ b_n = ...
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0answers
4 views

Least square estimator: $N( \beta x_i, \sigma^2)$

Let $ Y_1,...,Y_n$ be i.i.d $N(\beta x_i, \sigma^2) $ with known $ x_i's$. It is asked to find the Mean Squared Estimator for $\beta.$ I didn't understandmuch about this method of pbtaining an ...
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1answer
23 views

Expected Residual lifetime

I have a 2 part question. I was able to figure out part 1. I need some help with part 2. I will write out part 1 (and my solution) for completion. Let $T$ be a continuous survival time with survival ...
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1answer
68 views

Exam exercise on sequence $a_n = \sin(n)$ [duplicate]

Prove that the sequence $a_n = \sin(n)$ cannot converge when $n \rightarrow \infty $ I tried to find two subsequences that converge to different values but I am having trouble with the fact that $n ...
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2answers
82 views

Proving $(1-x)^n \geq 1 - nx $

$(1-x)^n \geq 1 - nx\,\, $ If i expand the left side of the inequality with the binomianl coefficient formula I obtain: $1-nx + {n \choose 2}x^2 - {n \choose 3}x^3 ... $ now I see where the $1-nx$ ...
2
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1answer
58 views

Uniform integrability of a sequence of functions.

I am considering the following sequence of functions. I think it converges pointwise to $0$ because the intervals in the domain in which the nth function is greater than zero eventually shrink and ...
4
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1answer
58 views

On a recursive sequence (exercise 8.9 Apostol)

The exercise states: show convergence of the sequence ${a_n}$ knowing that: $$|a_n| \le 2, \ \ \ |a_{n+2}-a_{n+1}| \le \frac{1}{8}|a_{n+1}^2 - a_{n}^2|.$$ The solution states: $$|a_{n+2}-a_{n+1}| ...