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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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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1answer
51 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
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0answers
107 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
42 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
36 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
82 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
41 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
122 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
42 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} ...
1
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2answers
123 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
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2answers
40 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
30 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
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2answers
64 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
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2answers
161 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
43 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
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1answer
21 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?
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1answer
174 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
37 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
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2answers
66 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
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1answer
60 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
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2answers
99 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 ...
2
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1answer
50 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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0answers
51 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
22 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
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1answer
83 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 ...
2
votes
2answers
37 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
60 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
7 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 ...
0
votes
1answer
36 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 ...
4
votes
1answer
95 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
89 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
120 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
votes
1answer
76 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}| ...
3
votes
1answer
105 views

exam exercise on Series problem.

The exercise states: Does the series $$\sum\limits_{n=1}^\infty \int_{0}^1 \frac{x^n dx}{x+1}$$ converge? The solution states as the first step: $$I_n =\int_{0}^1 \frac{x^n dx}{x+1} $$ then $ ...
2
votes
1answer
44 views

Testing convergence of the series of $n^p((n-1)^{-1/2}-n^{-1/2})$

Exercise 8.15 (l) of Analysis by Apostol states: Test for convergence: $$\sum\limits_{n = 2}^\infty n^p\left(\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt{n}} \right)$$ The solution I have states as the first ...
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2answers
179 views

Books/subjects for proof practice

So I want to practice writing proofs. I've studied general proof-writing but now I want to learn how to apply that to mathematics. From what I understand, the best and most accessible subjects for ...
2
votes
1answer
123 views

$[0,1]^{[0,1]}$ is separable

This is from Dudley´s book: Let $I:=[0,1]$ with the usual topology. Let $I^I$the set of all the functions from $I$ to $I$ with the product topology. a) $I^I$ is separable. Hint: Consider function ...
0
votes
1answer
30 views

Showing a set is closed, question from real analysis

Let $\mathbb{X}=\{1,2,3\}$ and let $\mathbb{P}$ be the set of all probabilities on $\mathbb{X}$. $\,\,$ Let, $V:\mathbb{P}\to \mathbb{R}$ be defined as $V(p)=(1+p_1)^2+(2+p_2)^2+(3+p_3)^2$. Show that ...
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2answers
42 views

How many ways there are to divide $5$-element set to at mot three sets?

How many ways there are to divide $5$-element set to at most three subsets? If I am right, then I have following subsets: 1 subset: containing five elements (that is 1 possibility) 2 subsets: 1 ...
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1answer
42 views

Find inverse of $I+\mathbf{ab}^\intercal$

Could you guys give me some hints on this homework? Find inverse of $\mathbf{I} + \mathbf{ab}^\intercal$. Hint: try to form $c\mathbf{I} + d\mathbf{ab}^\intercal$ and solve for $c,d$. What happens ...
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3answers
23 views

Simple Conditional Variance Proof Question

So I have the following best linear predictor: $y_{t+1} = a + b y_t + v_t $ , where $b$ is a measure of persistence and $v_t$ is noise (independent of $y_t$). Variance is persistent across ...
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4answers
166 views

Is there a closed form expression for $\int_{- \infty}^\infty \int_{-\infty}^y \frac{1}{2 \pi} e^{-(1/2) ( x^2+y^2 )} \mathrm{d}x\,\mathrm{d}y$?

I have been trying to evaluate the integral: $$\int_{- \infty}^\infty \int_{-\infty}^y \frac{1}{2 \pi} e^{-(1/2) ( x^2+y^2 )}\mathrm {d}x\,\mathrm{d}y$$ I know of course that the integral equals ...
3
votes
1answer
105 views

Introductory texts in abstract algebra, and game theory taking non-standard approaches

I like to see subjects from different angles. For example in linear algebra I'm reading through Axler's text (which takes a proof based approach for math students), but I'm also checking out a text on ...
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1answer
102 views

What is the (rigorous) reason that the derivative of $|x|$ does not exist at $x=0$?

Let $g=|x|$. Then, the derivative at $c=0$ is given by: $$ g'(0) = \lim_{x \to 0} \frac{|x|}{x} $$ which is either $+1$ if $x$ comes from the positive $x$-axis or $-1$ if $x$ comes from the negative ...
2
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0answers
194 views

A Compact Real Analysis book for a graduate student, who is short of time.

I am a Phd student in Computer Science and I want to focus on Machine Learning, especially on statistical methods. My problem is, I always keep hitting the wall when it comes to studying underlying ...
1
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1answer
29 views

Proof check from basic set theory

I wonder if my proof is detailed enough. $f:X\to Y.$ To be proved: $f^{-1}(\bigcap_{\alpha}E{\alpha})=\bigcap_{\alpha}(f^{-1}E_{\alpha}) $ $$\,$$ So, here goes ...
0
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1answer
54 views

On complex numbers and absolute values

Exercise 1.31 of Analysis by Apostol states: Given three complex numbers $z_1,z_2,z_3$ such that $|z_1| = |z_2| = |z_3| = 1$ and $z_1 + z_2 +z_3 = 0$. Show that these numbers are vertices of an ...
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0answers
136 views

Suggested book for self study.

I have a degree in Financial Risk Management, and did 4 semesters of calculus and analysis(but that was about 10 years back), with most of my other efforts going toward Mathematical Statistics and ...
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2answers
67 views

One Question on Law of Total Probability

Let $(X_n)$ with $n \in \mathbb N_0$ be a discrete martingale. Then I read the following identity which is said to be derived from the law of total probability. $$ \mathbb EX_m = \left( \sum_{n=0}^m ...
1
vote
1answer
46 views

Intuition underlying stopped martingales

Let $X$ be a martingale and $T$ a stopping time. Define the stopped martingale $X_{\min\{T,n\}}$. What is the intuition underlying this process? It is quite confusing here. $X$ is random and $T$ is ...