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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54 views

Prove that OLS estimator of the intercept has minimum variance

Let $$y_i=B_0+B_1X_i+\epsilon_i$$ where $\epsilon_i\sim > N(0,\sigma^2)$. Find the least squares estimator of $B_0$ and show that it is unbiased and has minimum variance. I will not write in ...
1
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1answer
71 views

New to Abstract Algebra, need guidance [closed]

I am new to Abstract Algebra. How should I begin learning it. On the face if it, it doesn't look easy to me.I have bought the book by Prof. Gallian. Is there any other book or any videos which I can ...
2
votes
2answers
53 views

Convergence of fixed point iteration when $g'(p)=1$.

I am dealing with a function $f(x)=e^{-\frac{1}{x^2}}$, which has a root $p$ of infinite multiplicity at 0. I am struggling with the convergence rate of the resulting standard Newton fixed point ...
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1answer
38 views

Book recommendations for learning financial/business mathematics.

Does anyone know a book which covers topics on: Simple interest Compound interest Equations of equivalent values Nominal rate, effective rate and equivalent rate Annuities ...
2
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0answers
33 views

What is the link/ relationship and difference between probability measure, Bernoulli measure, Lebesgue measure, Borel measure and Hausdorff measure

I am having difficulty in understanding what is the difference between probability measure and Bernoulli measure. Is the latter used when the random variable has a Bernoulli distribution? What is its ...
3
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1answer
55 views

Most Expeditious Way of Adding Consecutive Composite Numbers

In helping my 10-year old son with a homework problem that he was trying to solve by rote, I found myself resorting to arithmetic series. In short, I was calculating the arithmetic series from 12 to ...
0
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1answer
24 views

Jar has 10 dimes/12 nickels. After draw1, The coin that is drawn first is added back + 1 more of the same type. Find P(Dime1|Dime2)

To clarify, Find the probability that the first coin was a dime given the 2nd was also a dime. I'm very sorry for the extended title, but it says to be very specific. I think the tree diagram ...
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0answers
35 views

Joint and marginal pmf

A class of n students takes a test consisting of m questions. Suppose that student i submitted answers to the first m questions The grader randomly picks one answer, call it (I, J), where I is the ...
1
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1answer
43 views

Natural Isomorphism between $T_1^1(V)$ and End$(V)$

I'm a little stuck on showing that there is a natural isomorphism between the $\mathbb{R}$ vector space of $(1,1)$ tensors, and the $\mathbb{R}$ space of of linear maps $T:V\to V$. The hint is define ...
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1answer
36 views

Proof with stationary distribution

Let $\pi(k)$ the stationary distribution of the Markov Chain. Show that if $$p_{ij}^{(n)}\geq\varepsilon$$ for some $i,j,n,\varepsilon$ then $$\pi(j)\geq \varepsilon \pi(i)$$ I'm litle lost ...
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0answers
38 views

Markov Chain in urn with replacement

Consider a green ball and a yellow distributed in two urnas.Em each step, a ball is selected at random, then if that ball is green it changes of urn with probability $1/4$, and if the ball is ...
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0answers
37 views

Reversibility and stationary probability

Let $X_n$ a markov chain with transition matrix given by $$\begin{bmatrix}0.7&0.3&0\\0.2&0.7&0.1\\0.4&0.1&0.5\end{bmatrix}$$ i) Find the stationary probability ...
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0answers
49 views

Waiting times probability

Each entering customer must be served first by server $1$, then by server $2$, and finally by server $3$. The amount of time it takes to be served by server i is an exponential random variable ...
0
votes
1answer
15 views

Non-singular matrices - properties

Let $K=AA'$ where $A'$ is the transpose of $A$. $A$ is non-singular. Prove that $K\gt0$, or that all elements in $K$ are strictly non-zero. Not sure where to begin with this. I know that if $A$ is ...
6
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0answers
80 views

Concrete examples and computations in differential geometry

I've been studying differential geometry by myself for some time now. I studied a fair amount of the basic general theory and gone through a lot of the exercises from several textbooks. Lately I ...
1
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2answers
103 views

Proof that if $\phi \in \mathbb{R}^X$ is continuous, then $\{ x \mid \phi(x) \geq \alpha \}$ is closed.

Recently, having realized I did not properly internalize it (shame on me!), I went back to the definition of continuity in metric spaces and I found a proposition for which I was looking for a proof. ...
3
votes
2answers
59 views

what is a curve ? Is the concept of derivative limited to curves only?

I am trying to understand derivative and I want to know intuitive and rigorous definitions for a curve and if derivative is lmited only to curves or not..
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1answer
28 views

Support of random variable is a closed set

Let the support $S$ of a distribution function $F$ be $$S = \left\{x: F(x+\epsilon)-F(x-\epsilon) > 0, \forall \epsilon>0\right\}$$ I want to show that this is a closed set. In case anyone ...
5
votes
3answers
79 views

Advanced complex function theory book recommendation

I would like to have some recommendations in order to self study the above topic. I have already studied some complex function theory, covering some of the more 'classical' theorems (the Bloch-Landau ...
2
votes
2answers
59 views

Let $X_t$ and $Y_t$ Poisson Process

Let $X_t$ and $Y_t$ be two independent Poisson Process with rate parameters $\lambda_1$ and $\lambda_2$ respectively, measuring the number of customers arriving in stores 1 and 2. a)What ...
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votes
1answer
208 views

Fair coin tosses until until two consecutive heads or two consecutive tails appear

A fair coin is tossed repeatedly and independently until two consecutive heads or two consecutive tails appear. Find the PMF, the expected value, and the variance of the number of tosses. Let X be a ...
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4answers
80 views

What does $a\mid p$ mean?

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0answers
13 views

Polar plane spiral repitions

I'm just starting out teaching my self about the polar plane using tools like Desmos and have been wondering: When graphing an equation in the polar plane, does it extend forever? All the tools ...
0
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1answer
21 views

Rank of a product of a positive definite and a rank $2$ matrix.

If we assume that $\mathbf{U}$ is a matrix $n\times2$, with rank $2$ (two independent columns) and $\mathbf{A}$ is a positive definite matrix of order $n$, what would be the best way to see that the ...
13
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6answers
1k views

How do I teach university level mathematics to myself? [closed]

So here I go, I have enrolled myself in maths major this year but due to less marks in SSC I couldn't secure admission in a good university so I have to take admission wherever I could get with my ...
1
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1answer
29 views

How does this function behave in terms of $\gamma$ when we take $\lim_{b \to \infty}$

$$\mu_n = \frac{(b^{n-\gamma+1}-a^{n-\gamma+1})(-\gamma+1)}{(b^{-\gamma+1}-a^{-\gamma+1})(b-\gamma+1)}$$ I'm interested in how this behaves as $\gamma$ changes. We can assume $\gamma > 1$. I've ...
3
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0answers
83 views

What's the right way to read Princeton companion to mathematics?(for non-mathematician)

I am thinking of ways,how one(who is not a mathematician, but wants to know what's going on in the field of mathematics) can properly read Princeton companion to mathematics to make sense of it. I ...
3
votes
1answer
53 views

Being ready to tackle the math courses in my CS program

Here's my (long) story cut short. I was awful at math in high school. I did 4 years in the service and now I'm going to start college in just a few weeks. I am really nervous because I will have 5 ...
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0answers
56 views

I want to learn how to learn more. [closed]

Sorry this isn't really a directly related mathematics question but I've been observing the kids in my school and I'm just amazed. They are in Grade 11 and are doing calculus and all these high-order ...
1
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0answers
105 views

Solutions for “What is Mathematics?” by Richard Courant

I am currently reading through Courant's "What is Mathematics?" Most of the time I am not taking the exercises too seriously, given that I am reading this for pleasure, and that often my solutions are ...
1
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2answers
50 views

Question about Quantifiers

I've been reading Velleman's How to Prove It and it says that the $\forall x$ in$\forall x P(x) \rightarrow Q(x)$ only applies to $P(x)$ unless there is a parenthesis $\forall x (P(x) \rightarrow ...
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2answers
34 views

Problematic exercise on alternate way of expressing random variables

I found this exercise on a book on probability theory, and I find it problematic. Let $(X, \Sigma, p)$ be a probability space, $\mathcal{A} \subseteq \Sigma$ a finite partition of $X$; and $\phi ...
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3answers
86 views

What is the equivalent of musical ear training with regards to studying mathematics

When one aspires to be a professional musician, it is made clear that ear training is a very valuable skill that must be cultivated on a daily basis. The student is advised to put in the time and ...
3
votes
1answer
109 views

Measure from a “distribution function” and integrate with respect the associated measure

Hi this is an exercise from other old exam. I'd like to know if my attempt is correct. I'd really appreciate your help or your suggestions to improve my arguments and also recommendation to similar ...
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votes
1answer
30 views

a question related least common multiple [closed]

$ 1, 3, x, 15, y, 75$ These above six numbers are divisors of a number $z$, and these numbers are ordered from the least one to the highest one. How to find numbers $x, y, z$?
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1answer
88 views

textbook for self studying geometry

Looking for recommended readings in geometry for self study. I am planning to get this book for self study. Geometry: A High School Course
3
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1answer
40 views

Explicit construction of a sigma algebra that makes a simple function measurable

This is a question from an old exam. I'd like to see if my answer is correct. I'd appreciate any suggestion. Thanks :) Let $f: (\mathbb{R},\mathscr{S}) \to (\mathbb{R}, \mathscr{B}(\mathbb{R}))$ ...
0
votes
1answer
74 views

Probability of hiting the target

If the probability of hitting a target is 1/5, and ten shots are fired independently, what is the probability that the target is hit at least twice? What is the conditional probability that the target ...
32
votes
9answers
3k views

How to self learn mathematics if you can't buy the books? [closed]

I'm trying to learn more mathematics, especially number theory and abstract algebra, with my goal being category theory, however, whenever I look for book suggestions here, most guys will link to ...
1
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0answers
29 views

Condtions to make a semiring a base for a topological space

In the book "Principles of Real Analysis" by Aliprantis and Burkinshaw I found the following exercise: Let $\mathcal{S}$ be a semiring of subsets of a nonempty set $X$. What additional ...
0
votes
1answer
15 views

Combinatorics problems: termination at rth step

In Feller's book of probability exixt such formulas: a)Placing balls untill for the first time a ball is placed into a cell already occupied: The probability of the process termitating at the rth ...
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votes
2answers
67 views

Span of two vectors in $\mathbb{R}^2$ [closed]

The span of two vectors in $\mathbb{R}^2$ neither of which is zero vector, and which are not parallel, is- a point. line in $\mathbb{R}^2$ not running through origin. line in $\mathbb{R}^2$ running ...
0
votes
3answers
96 views

Easiest (most forgiving?) way to learn category theory?

I'm not a maths student, but I've read a bit on category theory and I'd love to learn about it. Is there any book that's simple enough for a busy student to pick up and learn the fundamentals? What ...
0
votes
1answer
31 views

Find the sum of this series $\sum_{k=1}^{\infty} \frac{(2-x)^k}{2^k*k}$ How to find the integration $C$?

Hello I have the following series : $$\sum_{k=1}^{\infty} \frac{(2-x)^k}{2^k*k}$$ I found that the series convergents for $0<x \leq 4$ I managed to reach that $$f(x)=-ln(x)+C$$ But for some ...
2
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0answers
76 views

Rationale behind construction of measure theory from semirings

I am studying a book (Aliprantis & Burkinshaw, "Principles of Real Analysis") that, in order to introduce the concept of measure, starts from semiring. In particular the authors state that: ...
0
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1answer
86 views

Reference request for Stochastic Processes in general

I'm studying stochastic processes through the book "Introduction to Stochastic Processes, Gregory F Lawler". Is there any significant difference between "Stochastic processes, Sheldon Ross" and ...
2
votes
1answer
162 views

How to evaluate the integral $\int_0^{\infty}[I_{(0,2)}(z)]\frac{(n-1)(y-z)^{n-2}}{y^{n-1}}dy$

I am doing a statistical calculation from a statistical exercise but get stuck at the following integral. $$\int_0^{\infty}[I_{(0,2)}(z)] \frac{(n-1)(y-z)^{n-2}}{y^{n-1}}dy$$ $0<z<y$ ...
0
votes
1answer
28 views

Is small triangle is similar to big triangle

A triangle, $ABC$, has point, $M_1$, $M_2$, $M_3$, where $M_1$ is the mid point of Line $AB$, $M_2$ is the mid point of Line $BC$, and $M_3$ is the mid point of Line $AC$. A smaller triangle, ...
1
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2answers
71 views

Proof in ruin player problem

Let $M_i$ the average number of matches until the player, or lose all, or wins the capital $N$ as it began with the capital $i$. Show that $$M_i=i(N-1);p=\frac{1}{2}$$ ...
2
votes
1answer
20 views

Image of matrix $\int_0^t e^{sA}BB^T e^{sA^T} ds$

Let $A \in R^{n \times n}$ and $B \in R^{n \times m}$. Define $$Q_t = \int_{0}^t e^{sA}BB^T e^{sA^T} ds$$ Suppose that $x \in \text{Im } Q_t$, ie, $\exists \eta \in R^n$ such that $$x = Q_t \eta$$ ...