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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3
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1answer
89 views

How to avoid losing the woods for the trees in daily study/lecture time

When facing to some new material in mathematics, I feel easily to be overwhelmed by lots of details with losing the woods for the trees. So is there some good strategy to study the materials ...
4
votes
1answer
211 views

Uncountable Dense Linear Orders

Is there an example of two uncountable equipollent dense linear orders without endpoints that don't satisfy the same first order properties? Or is it true that two uncountable equipollent dense linear ...
0
votes
0answers
135 views

On what value of $f(X)$ minimizes $E[(Y-f(X))^2|X]$

$X$ and $Y$ are random variables. The question is: what value of $f(X)$ minimizes $E[(Y-f(X))^2|X]$. I am pretty sure I have found the solution to this problem by writing: $$E[(Y-f(X)-E[X|Y] +E[X|Y] ...
4
votes
0answers
93 views

How to select good exercises?

I'm studying on Rudin "Principles of of Mathematical Analysis" which I begin to find as a good and complete reference. I wonder how many exercises shall I do at the end of each chapter ? In case of ...
3
votes
2answers
101 views

Determining final and initial object in a certain category

I am reading Paolo Aluffi's greatly entertaining book "Algebra: Chapter $0$" and I got stuck on some excercises dealing with universal properties. Let $C$ be a category, and let $A$ and $B$ be two ...
1
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1answer
61 views

Normal Distribution and Iterated Logarithm

Let $X_n$ be independent $N(0, \sigma^2)$-distributed random variables with partial sum $S_n := \sum_{k=1}^n X_k$, $n \geq 1$. Then I read the following results. $$ \sum_{k = 1}^n \mathbb P (S_n > ...
0
votes
3answers
36 views

Limit with terms very similar to those that should give an exponential function

I have been trying to solve the following limit but am completely stuck. $$\lim_{\alpha \rightarrow \infty} 1-\left( \frac{y+\alpha}{\alpha-1} \right)^{-\alpha}$$ I have tried inverting the ratio ...
-2
votes
1answer
507 views

Posterior Probability

a) Your initial belief is that a defendant in a court case is guilty with probability 0.5. A witness comes forward claiming he saw the defendant committed the crime. You know the witness is not ...
0
votes
1answer
37 views

Confusion about the associative property and the mechanics of Parenthesis

This is a follow up question on my earlier post (Updated): Showing that a set $M$ with two elements classifies as a field. I feel this post is necessary because I realize that what confuses me is how ...
1
vote
1answer
42 views

Simple probability question

Question: In class of 125 students, in examination 70 students passed in mathematics and 55 students passed in statistics and 30 passed in both the subject. Find the probability of the event where ...
0
votes
1answer
40 views

(Updated): Showing that a set $M$ with two elements classifies as a field

My question is more conceptual, so I will come straight to the exercise: Exercise: Let $M= \lbrace g,u \rbrace $ be a Set. On $M$ the Addition and the Multiplication is given by: \begin{align} ...
1
vote
4answers
184 views

Constructive proof for existence of integer part of real number

I try to prove de following exercise of my analysis textbook. Show that for every real number $x$ there is exactly one integer $N$ such that $N \le x < N + 1$. I have been finding a ...
2
votes
8answers
126 views

Proof that $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$

Any hints on how I can prove the inequality: $$|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$$ Thank you.
0
votes
3answers
41 views

Is this derivative correct?

I'm newbie at Calculus, so I'm doing some exercises of derivates, I know by the formula: $f(x) = \sqrt u$ $\frac {df(x)}{dx} = \frac{u'}{2 \sqrt u}$ that the derivate of the next function is: ...
1
vote
1answer
80 views

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 ...
0
votes
1answer
47 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 ...
0
votes
1answer
604 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
vote
1answer
46 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, ...
3
votes
3answers
565 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 ...
2
votes
1answer
93 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 ...
2
votes
2answers
45 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 $h$ ...
1
vote
1answer
72 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 ...
5
votes
1answer
109 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 ...
1
vote
1answer
545 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 ...
0
votes
1answer
33 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 ...
1
vote
2answers
140 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 ...
2
votes
1answer
52 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} ...
1
vote
1answer
61 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 ...
4
votes
0answers
141 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 ...
1
vote
1answer
89 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> ...
1
vote
1answer
38 views

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

How to show the following result about series? Thank you! Convergent Series: $$\sum_n \frac{1}{n^q}, \ \ q>1$$
0
votes
1answer
111 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)
0
votes
3answers
43 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 ...
1
vote
2answers
178 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 ...
1
vote
2answers
181 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 ...
0
votes
2answers
44 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 ...
1
vote
0answers
34 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 ...
0
votes
2answers
81 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 ...
2
votes
2answers
167 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
votes
1answer
63 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 ...
0
votes
1answer
22 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?
3
votes
2answers
2k 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
votes
1answer
45 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 ...
0
votes
2answers
68 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 ...
0
votes
1answer
62 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 $y_{ix}$ ...
0
votes
2answers
110 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 ...
3
votes
1answer
57 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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vote
0answers
59 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 $\frac{a_{n+1}}{a_n}$ ...
1
vote
1answer
25 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 ...
1
vote
1answer
90 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 ...