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.

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efficiency of estimator, $\bar (\frac{1}{X^2})$ vs $\frac{1}{\bar X^2}$ vs $\frac{1}{\bar (X^2)}$

I was studying point estimator, and I tried to compare the variances of the estimators to find out which one is more efficient. (Hogg, Tanis "Probability and Statistical Inference" Ch.6) It ...
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1 vote
2 answers
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For a set of n Exponential iid RV, what is the probability that the maximum exceeds the sum of the others?

My question stems from self-study of question 91 in page 372 of Ross's Introduction to probability models, 12th edition. The answer is given in the textbook but I am trying to understand a specific ...
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1 answer
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Proof of convergence of random variables in $L^p$ via convergence in probability and uniform integrability.

Consider the following proposition. Part (i) i have no problem with. Its the proof of part (ii) that (because of my lack of knowledge of advanced measure theory) am having trouble in understanding. ...
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47 views

How to know what to study based on past examinations?

TL;DR: Tell me which topics should i study the most, based on this three tests: Mathematics (A): 2020 2019 2018 This question may sound a bit weird, since the natural answer would be "study ...
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best textbooks for difficult specialist maths questions in year 11

I am currently studying year 11 specialist maths in Western Australia and I am doing the O.T. Lee textbook, however, I am looking for harder questions that will properly challenge my conceptual ...
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Miscellaneous questions about Lebesgue measurable set and Lebesgue measure [closed]

(Definition of Lebesgue measurable set and measure) If a set $\mathcal{M}$ and a measure $\mu$ satisfy i) $\mathcal{M} \supset \mathcal{B}(\mathbb{R}^n)$, ii) any rectangle subset of $\mathbb{R}^n$ ...
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1 vote
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Given $|u-y|<\delta$ for some $\delta>0$, $u,y\in\mathbb{R}$ and $n\in\mathbb{N}$, proving $|u^n-y^n|<\epsilon$ for some $\epsilon>0$ by induction.

Here is the full version of the question: Given $y\in\mathbb{R}$, $n\in\mathbb{N}$, and $\epsilon>0$, show that for some $\delta > 0$, if $u\in\mathbb{R}$ and $|u−y| < \delta$ then $|u^n −y^n|...
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Why do we care whether the support of a function is compact or not?

This is a question for self-learning. I am too confused by the text to formulate a well-defined question now. I am reading analysis of functions, and confused by the motivations of some theorems and ...
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26 views

Describing the set of elements that are in at least one of the sets A or B

Textbook problem: Use unions, intersections, and complements to express the set of elements that are in at least one of the sets $A$ or $B$, where $A$ and $B$ are subsets of the set $\Omega$. My ...
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2 votes
1 answer
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Prerequisite mathematics for nonlinear systems [closed]

I have a background in electrical engineering and linear control systems. I want to learn nonlinear systems. There is a book Nonlinear systems by Hassan K. Khalil. The book has a lot of advanced ...
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Expressing the multiplication of two Dedekind cuts in $\mathbb{Q}$ (Pugh)

Let $x=A|B,x′=A′|B′$ be cuts in $\mathbb{Q}$. Why do we not define $x·x′ = (A·A′)| \text{rest of}\,\mathbb{Q}$? My intuition was to define $A=A'=\{r\in\mathbb{Q}|r<0\vee r^2<2\}$, as both $x$ ...
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1 vote
1 answer
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Explanation for the Euclid Extended Algorithm

Please this is how the code for the extended-euclid algorithm was implemented in the book Introduction to Algorithm (Chapter 31 page 937 EXTENDED-EUCLID(a,b) if b == 0 return (a,1,0) else (d_1,x_1,y_1)...
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Plotting nullclines for a planar system of non-linear ODEs

A past exam paper I'm working on (Mathematics BSc, second-year module in differential equations, unpublished) has a question to plot the nullcline diagram of the system of ODEs, \begin{align} & \...
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calculate the conditional expectation $E[D |s] =?$

I know that Given the joint normality $(D, \epsilon)$, it is defined that $s = D + \epsilon$ where $\epsilon \sim N(0,\sigma_{\epsilon}^2)$ and $D \sim N(\bar{D}, \sigma^2)$ $D$ and $\epsilon$ are ...
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2 votes
2 answers
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Composition of Riemann-integrable and increasing functions.

This is exercise 7.3.3 from Abbot's Understanding analysis. The section is Integrating functions with discontinuities. I am struggling with this exercise. I can't either come up with any simple ...
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Finding the mean of $\big(\sum^{n}_{j=1}{X_{ij}}\big)^2$ where $X_{ij}$ is an indicator variable

Let $n_i$ be the random variable denoting the number of elements placed in bucket $B[i]$ and let $X_{ij}$ be the indicator variable $\mathbb{I}\{A[j] \text{ falls in bucket } i\}$. Thus, for each $i$, ...
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Proof checking: $B\subseteq A$ implies $\sup B \leq \sup A$

Given $B \subseteq A$. Assume $\sup B > \sup A$. Now $\forall \epsilon >0$ $\exists b\ \in B (b > \sup B - \epsilon)$. Take $\epsilon = \sup B- \sup A$. That is $\exists b\ \in B (b > \...
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2 votes
2 answers
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How many of each specific coins are in a bottle?

Question Suppose you have a bottle that contains exactly twenty-two U.S. coins.These coins only consist of nickels(\$0.05), dimes(\$0.10), and quarters(\$0.25). In addition to the types of coins, you ...
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1 vote
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Please check my proof from supremum infimum topic

Q : Let A be bounded below and define B$ = \{b \in \mathbb{R} $:b is lower bound for A $\}$ To Show : supB=infA Proof : Since A is bounded below, lets assume that $a$ is lower bound for A. so $a\in B$....
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Self Study: Proof $Cov(X, E[X|Y]) = Var(E[X|Y])$

Like the title, here is what i've done so far: \begin{align*} Cov(X, E[X|Y]) &= E(XE[X|Y]) - E[X]E[E[X|Y]] \\ &= E(XE[X|Y]) - E[E[X|Y]]E[E[X|Y]] \\ &= E(XE[X|Y]) - E[E[X|Y]]^2 \\ \end{...
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How is $e^{\frac{-k(k-1)}{2n}} \leq \frac{1}{2}$

I was learning from the book Introduction to Algorithms (section 5.4.1) and came across the birthday paradox. This was the solution I saw: $Pr\{B_k\} = 1. (1 -\frac{1}{n})(1-\frac{2}{n})...(1 - \frac{...
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2 answers
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Turning a pole/ladder horizontally at a corner

I've been attempting to solve the following problem: A lane runs perpendicular to a road 64 ft wide. If it is just possible to carry a pole 125 ft long from the road into the lane, keeping it ...
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64 views

Obscure Results and Areas of Mathematics

My degree is in economics although I did relevant mathematics in college and used algebraic topology in my thesis. Over the Pandemic I've been reading a lot of mathematics that I didn't get a chance ...
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Calibrating a Discrete Autoregressive Model using the Method of Moments

Consider the following discrete autoregressive $\epsilon_t$, where $\epsilon_t \in (\pm 1 ) \forall \ t \geq 1$. We think of $\epsilon_t$ as the child of a previous sign at time $t-l$, where $l$ is ...
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-1 votes
0 answers
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Calculation of the given Expected value

$$E[ -a(z(D - \bar{D}))]$$ I would like to calculate this expectation. This expectation is over D. D is normal distribution and $D \sim N(\bar{D}, \sigma^2)$ and z is normal distribution and $z \sim N(...
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How is $(2^a)^{\lg n} = n^a$? [closed]

I was learning from Introduction to Algorithms (Chapter 3 under the topic “Logarithms”) and came across this expression. $$ \lim_{ n \to 0 }{\frac{\lg^b n}{(2^a)^{\lg n}}} = \lim_{n \to 0} \frac{...
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1 vote
1 answer
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Discrete time forward rate

I'm reading Jarrow and Turnbull (1997) . They defined p(t,T) as the time t price of a default free zero coupon bond paying a sure dollar at time T where $0\le t \le T$ (in year) . They also defined ...
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How can I get more comfortable working with integrals and bounds in analytic number theory?

I've started self-studying analytic number theory from these lecture notes (I am currently attempting exercises from Chapter 2), and even though I enjoy the learning process, my main difficulty seems ...
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1 vote
1 answer
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Convergence in Probability of Positive Random Variables

We have to show that for a positive sequence of random variables $X_n \geq 0$ and a random variable $X$ s.th. $X_n \overset{\mathbb{P}}{\rightarrow}X$ it follows that $\mathbb{P}(X \geq 0) = 1$. ...
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1 vote
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Proving the Weierstrass Approximation Theorem using Polygonal functions

This is exercise 6.7.8. from Abbott's Understanding analysis 2nd ed. As the question title says, this section is about the Weierstrass Approximation Theorem (WAT). Abbott guides us to prove WAT ...
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-3 votes
1 answer
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Proof that (-1)x + x = 0 [closed]

I was going through "The Art of Problem Solving - Prealgebra" textbook and I noticed that one of the examples include a proof for (-1)x + x = 0 My solution differs from their proof, but here'...
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5 votes
1 answer
126 views

Approximating the absolute value function with a polynomial, starting from the Taylor series of $\sqrt{1-x}$

This is Exercise 6.7.7 from Abbott's Understanding analysis. For context, the section is about the Weierstrass approximation theorem, but this is a step towards proving it, so we cannot use WAT to ...
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converging sequences in new topology

Suppose $(X,T_X)$ is a topological space and $\infty_X \notin X$. Write $X^* = X \cup \{\infty_X\}$ and suppose the open sets of $X^*$ the empty set and the union of an open set in $X$ and the point $\...
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How to be better at Mathematics(Algebra mostly), Probability and Statistics?

Can anyone suggest how to get better at solving Maths, specifically Probability and Statistics? I'm a first-year Engineering student in India preparing for an entrance exam to pursue a master's in ...
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-1 votes
1 answer
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Soft-Question: Is it possible to visualize, or make concrete, progressively more abstract mathematics? Are there mathematicians who can?

This is my first post! I'm really not good at math, and I'm trying to re-learn on Khan Academy. As I'm progressing, this question came up for me. Learning math concepts that I can visualize in my mind ...
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How do one systematically self-learn math?

If one wants to self-study advanced undergraduate courses or basic graduate courses, how to promise that the knowledge is "systematically" organized? I don't even know how to define "...
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0 votes
0 answers
23 views

Proof Intersection with Pairwise Disjoint Sets, Correct?

Statement: Let $A, B_{i=1,...,n} \in \Omega$ with $B_i \cap B_j = \emptyset \ \forall i\neq j \in 1,...,n$, then: $(A \cap B_i) \cap (A \cap B_j) = \emptyset$. Proof (attempt): \begin{align*} B_i \cap ...
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Can you take out a common factor from AR coefficients in a time series process after differencing?

here Is a solution to one of the questions from the book "Time series analysis with applications in r", where we need to find expectation and variance of a process after first differencing. ...
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3 votes
1 answer
170 views

Taylor series of $\sqrt{1-x}$

This is problem 6.7.6 in Abbott's Understanding Analysis 2nd ed. The section is called The Weierstrass approximation theorem. a) Let $c_{n} = \frac{1\cdot 3\cdot 5\cdots (2n-1)}{2\cdot 4\cdot 6\cdots ...
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1 vote
1 answer
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Laplace transform of $f(t)= \frac{d_1}{d_2}\frac{1}{\sqrt{4 \pi Qt}}\frac{d_2-d_1}{t}\exp\left(-\frac{(d_2-d_1)^2}{4Qt}\right)$?

I came across the following Laplace transform of $f(t)$ in a journal article: $$f(t)= \frac{d_1}{d_2}\frac{1}{\sqrt{4 \pi Qt}}\frac{d_2-d_1}{t}\exp\left(-\frac{(d_2-d_1)^2}{4Qt}\right).$$ The solution ...
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1 vote
0 answers
27 views

Self-Study: Uncountably discontinouities in CDF despite $\mathbb{P}(X = x) = 0$

Let $X: \Omega \rightarrow \mathbb{R}^k$ be a random variable and let $F_X(x)$ be it's cumulative distribution function. In $\mathbb{R}^1$, $\mathbb{P}(X = x) = 0$ implies that $F_X(x)$ is continuous. ...
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5 votes
1 answer
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Every connected graph has a spanning subtree

I would like to verify my proof of this basic statement (also the terminology): Statement: Given a connected graph $G$, it always contains a spanning tree as a subgraph. I try to prove this by ...
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2 votes
0 answers
52 views

Is it worth learning measure theory and Lebesgue integration using the Daniell scheme and Shilov's book for a first timer self-studying?

I'm an undergraduate with somewhat little analysis background (one undergraduate course a year ago and a graduate complex analysis course this semester) who wishes to self-study advanced real analysis ...
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0 votes
1 answer
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Check the monotonicity of an inductive sequence

Given $$a_{n+1} = \sqrt{\dfrac{ab^2+a_n^2}{a+1}}\,\,\forall n$$ Where $a>0, 0< a_1< b, a=a_1$ I obtained the expression for terms $a_2^2, a_3^2, a_4^2,...,a_n^2$ and do the summation of $a_i^...
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1 vote
1 answer
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Does this proof of the Boundedness Theorem contain a mistake?

My course notes (mathematics BSc, second-year module in real analysis, unpublished) have a proof of the Boundedness Theorem which begins: But does that sequence work? Here's my reasoning. Let \begin{...
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  • 1,495
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1 answer
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Question in probability regarding bivariate normal distribution

Let $(X_1,X_2)$ follow bivariate normal distribution with: $\mathbb{E}(X_1)=\mathbb{E}(X_2)=0$ $Var(X_1)=1, \ Var(X_2)=2$. AND $\text{Corr}(X_1,X_2)=\frac{1}{2}$ Let $Y_i=e^{X_i}, \ i=1,2$. Calculate $...
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find the conditional distribution of multivariate normal using the following fact.

Consider $X = \begin{bmatrix} X_p \\ X_q \end{bmatrix} \sim N( \begin{bmatrix} \mu_p\\ \mu_q \end{bmatrix}, \begin{bmatrix} \Sigma_p,\Sigma_r\\ \Sigma_r^\top,\Sigma_q \end{bmatrix} ) $ where $\Sigma$ ...
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2 votes
1 answer
131 views

A good book on topology for self-study

What's a good book for self-study on a first topology course? I'm taking topology for the first time, and my professor isn't what you might call great, everything is utterly easy or trivial for him, ...
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1 vote
1 answer
39 views

Finding the set of reachable points given time and speed

The following problem is from The Method of Coordinates by I.M. Gelfand et al.: What is a good way to approach this? I provide my very rough sketch for part (a) below (which I am not terribly ...
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14 views

Show that the F statistic for dropping a single coefficient from a model is equal to the square of the corresponding z-score

This is the solution that I found So I have been trying to understand this solution and I am totally lost. Why is the numerator of F become a chi square? Same idea with the denominator. On top of that,...
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