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 fact that you're self-studying is what your question is _about_.

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Can anyone suggest a reference to learn about relative log-likelihood and likelihood intervals?

I want to understand how to calculate the 10% likelihood interval for a Poisson model of count data. It is an old assignment where they give you 20 counts, tell you it is a Poisson model and ask you ...
5
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2answers
70 views

How to minimize $x^2+4xy+5y^2-4x-6y+7$ without using calculus

I would like to find the smallest possible value of the function $$f(x,y)=x^2+4xy+5y^2-4x-6y+7$$ without taking any derivatives. My thoughts were to complete the square on both $x$ and $y$ and ...
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1answer
31 views

What is the number of words of length $h$ in a sequence of subsets of words?

Let $L=\{0,1\}^*$ (the set of binary words on $0$ and $1$), Given an integer $k$, and $S$ a finite subset of $L$ define recursively the following sequence of subsets of $L$: $$\begin{align} A_1 ...
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1answer
110 views

How to stay productive while you are studying math? [closed]

Not sure that this question is a good fit for this site, but I will try. When I am working through a chapter of a mathematical book first two hours are normally very productive (easily remember ...
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2answers
219 views

Prerequisits for Gauss-Green theorem

Consider the following theorem from the appendix C from Evans PDE book: I know about integration in $\mathbb{R}^n$ but not about how to make sense of the integrals on the right-hand side. As my ...
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14 views

Scheduling problem

Consider the following setting: $N$ jobs, each has a starting time, which is assumed to be a natural number and all N numbers are distinct, e.g., the 1st job has starting time at 5, the 2nd is 6, the ...
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12 views

Reference material on Alternating Minimization Algorithm

I am looking for some good reference material (book/paper) for learning Alternating Minimization Algorithm. Any recommendation from optimization experts will be much appreciated. Thank you.
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2answers
27 views

Question about an exercise from Feller

The following is an exercise from the classical textbook of Feller on probability theory. Four girls take turns at washing dishes. Out of the total of four breakages, three were caused by the ...
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4answers
71 views

Is $f\colon\mathbb{Z}\to\mathbb{Z}, f(x)=x^2$ injective? Surjective?

I would say no: $\text{Suppose } f(a)=f(b) \text{ then } a^2=b^2 \implies \pm a = \pm b \implies -a=b$. Or simply by counterexample: $f(-1)=f(1)$ Further, I would say it does not map $\mathbb{Z}$ ...
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1answer
57 views

Infinite horizon cost function

The following quote is from Bertsekas's Dynamic Programming and Optimal Control. I'm only looking for a nudge in the right direction as to how to interpret the following equations, particularly ...
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3answers
335 views

Equivalence Relation between Derivative Being Odd and Function Being Even

In the exercise, I am required to prove that $f'$ is odd $\iff$ $f$ is even Moving from right to left was pretty trivial, however, I couldn't move from left to right. Note that we can only use very ...
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2answers
22 views

If $A_n \downarrow A.$ then $A_1 - An \uparrow A_1 - A$? Set theory.

Let $A_1, A_2 , \dots$ be subsets of a set $\Omega$. If $ A_1 \subset A_2 \subset \dots$ and $\bigcup_{n = 1}^{\infty} A_n = A $ then we write $A_n \uparrow A.$ $ A_1 \supset A_2 \supset \dots$ and ...
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1answer
17 views

liminf and limsup in probability

Consider a sequence of random variable $(X_n)$. Prove the following inequality: $$ \mathbb P\left(\liminf\{X_n \leq x\}\right) \leq \liminf\mathbb P\left(\{X_n \leq x\}\right)\leq \limsup\mathbb ...
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1answer
34 views

Set Inclusion Properties

Consider a sequence of random variables $(X_n)$ converge almost surely to $X$. Define set $N:=\{\omega: X_n \to X \}^C$. Then it is claimed that we would have the following set inclusion properties. ...
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2answers
44 views

$4$ random digits, $2$ different ones.

I have been trying to solve the following problem What is the probability that among $4$ random digits, there appear exactly $2$ different ones? Two different digits means that there should be ...
2
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1answer
99 views

Proof that this specific function is measurable

Bounty Edit: Considering the nature of the problem at hand (i.e. proving that a specific function is measurable), I think this can be an easy but relevant problem. In particular, it is relevant to ...
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2answers
38 views

Prove cyclic group with one generator can have atmost 2 elements

Prove cyclic group with one generator can have atmost 2 elements . Attempt Consider a cyclic group generated by $a \neq e$ ie G = .So G is also generated by <$a^{-1}$> .Now Since it is given ...
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31 views

How to construct a two sided confidence interval?

A machine produces metal rods used in an automobile suspension system. A random sample of 15 rods is selected and the diameter is measured. The resulting data are shown below. 5.21 5.28 5.29 5.27 ...
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1answer
22 views

Consequence of linear combination in matrix .

If a column of a matrix is linear combination of another column, what are the consequences ? Several terminology coming into my mind to relate with this such as Rank of the matrix ; Determinant ...
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1answer
46 views

Showing that Determinant is a Volume Multiplier

I want to show using the change of change of variables theorem for (Riemann) integration that the determinant of a linear transformation $T$ is a scaling factor for the volume of a space. If $1_A$ is ...
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0answers
32 views

Convergence of Types Theorem

(Convergence of Types Theorem) Suppose that $F_n(u_nx+v_n) \Rightarrow F(x)$ and $F_n(a_nx+b_n) \Rightarrow G(x)$, where $u_n>0, a_n>0$ and $F$ an $G$ are non-degenerate. Then there exist ...
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1answer
67 views

$P( A\triangle B ) = 0 \Rightarrow P(A)=P(B)=P(A\cup B) + P(A \cap B) $

I want to prove the following statement; $$P( A\triangle B ) = 0 \Rightarrow P(A)=P(B)=P(A\cup B) \color{blue}{=} P(A \cap B) $$ What I did is that $$P(A\triangle B) ...
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3answers
57 views

$ P( A \mathrel{\triangle} B ) \le P(A \mathrel{\triangle} C) + P(B\mathrel{\triangle} C)$

Show that $$ P( A \mathrel{\triangle} B ) \le P(A \mathrel{\triangle} C) + P(B\mathrel{\triangle}C)$$ where $\mathrel{\triangle}$ indicates the symmetric difference I cannot write my idea, because ...
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1answer
28 views

Definition of Multiple .

Definition of multiple is : In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for ...
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3answers
90 views

If M,N are finite dimensional vector spaces with same dimension ,then if M is subset of N ,then M=N

If M,N are finite dimensiona;l vector spaces with same dimension then if M is subset of N ,then M=N I think i need to show that both vector spaces are spanned by same bases in order to do this or to ...
2
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0answers
61 views

Can anyone check if this correct?

Convert to spherical coordinates and evaluate:$$\iiint_{E}z(x^2+y^2+z^2)^{-3/2}dV$$ where E is the region satisfying the following inequalities:$$x^2+y^2+z^2\le16,z\ge 2$$ This is what i have done so ...
2
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1answer
36 views

Second moments from survival function

Let X be a non-negative continuous random variable with probability density function f(x). Let $$G(t) = \int_{t}^{\infty} f(x)dx$$ Show that$$E(X^{2}) = 2\int_{0}^{\infty} tG(t)dt$$ My thoughts: I ...
0
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1answer
33 views

limt of the function as $\mu\rightarrow\infty$ or $\mu\rightarrow-\infty$ .

$\lim_{\mu\rightarrow\infty}\frac{\exp(\bar x-\mu)^2}{(\bar x-\mu)}=? $ Also, $\lim_{\mu\rightarrow-\infty}\frac{\exp(\bar x-\mu)^2}{(\bar x-\mu)}=? $ I know, ...
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2answers
228 views

Is $g(x)=\log x$ convex function?

The graph of convex function is : In a book it is written that $g(x)=\log x$ is strictly convex function. So i searched for graph of $g(x)=\log x$ and found that Though it has been said that ...
2
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2answers
71 views

To prove any two basis of Finite Dimensional Vector Space have same number of elements

To prove any two basis of Finite Dimensional Vector Space have same number of elements If i take bases as $S_!$ = {$\alpha_!$ ,$\alpha_2$ ,....$\alpha_n$ } $S_2$ = {$\beta_!$,$\beta_2$ .... ...
2
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0answers
21 views

Uniform probability bound - checking my understanding

Let x and y be two independent random variables. What is the difference between (1) $P_x[\forall y, f(x,y) < \epsilon] >1- \delta$ (uniform bound), and (2) $\forall y, ...
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1answer
37 views

gluing together continuous functions

HI I was checking and old question here and I have troubles to proof the following: Proposition: Let $X$ be a space with subspaces $Y,A,B$ such that $X \backslash Y= A \sqcup B$ (disjoint union). Let ...
2
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1answer
77 views

To prove set is a group

Given a non empty set together with associative binary operation $*$ on $G$ such that $a*x=b$ and $y*a=b$ have solutions in $G$ for all $a,b$ in $G$ To prove it is a group Hints to get started ...
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1answer
17 views

Sample Points Calculation.

A coin is tossed three times. There are three sample points that i can get one head and two tail. I can count the sample points after writing the sample space, as ...
6
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1answer
181 views

Mathematical introduction to machine learning

At first glance, this is once again a reference request for "How to start machine learning". However, my mathematical background is relatively strong and I am looking for an introduction to machine ...
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1answer
23 views

To find right cosets of H in G where G=<a> and H=$<a^{2}>$ ,where o(G)=10

To find right cosets of H in G where G= and H=$<a^{2}>$ ,where o(G)=10 Since order of $G =10$ , so $a^{10}=e$ .We have $G= { a,a^{2},a^{3},a^{4},a^{5},a^{6},a^{7},a^{8},a^{9},e}$ and $H = ...
1
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1answer
68 views

If G has no non trivial subgroups ,then Show that G must be of prime order

If G has no non trivial subgroups ,then Show that G must be of prime order .This question is from Herstein Page 46 Question 3 . Attempt :- Let G has prime order(say p) .So by Lagrange theorem ...
2
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0answers
43 views

On the importance of the Riesz–Markov–Kakutani representation theorem.

I am following big Rudin and I have arrived at the representation theorem. Before doing the full long proof I would like to know what results are based on this theorem that for completeness I state ...
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4answers
264 views

To prove every element of G has finite order where

Let G be a group such that intersection of all its subgroups which are different from e is a subgroup different from e . To prove every element of G has finite order Hints to get started Thanks ...
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2answers
41 views

To prove $o(HK) = o(H)o(K)/o(H\cap K)$

Given that $H$ and $K$ are finite subgroups of $G$ of order $o(H)$ and $o(K)$, prove that $$o(HK) = \frac{o(H)\,o(K)}{o(H\cap K)}$$ I have proved for specific case when $H$ and $K$ have only ...
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0answers
27 views

How to show that a function is continuous in the topology of weak convergence

Let $\Omega$ be compact, and let $\omega^* \in \Omega$ be arbitrary. Let $\Delta (\Omega)$ denote the set of all probability measures over $\Omega$, and endow $\Delta ( \Omega)$ with the topology of ...
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3answers
87 views

Showing $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$

To show: $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$ Now, the equation $x^{2} =3$ has a solution in $\mathbb{R}$, but not in $\mathbb{Q}$. Hence they are not isomorphic to each other. Is ...
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1answer
33 views

how to get the second equation (related to summation)

$$V(Y) = \sum_{i=1}^N\sum_{j=1}^N [\frac{N^2}{n^2}] (Y_i-Y_j)^2 \frac{n(N-n)}{N(N-1)} $$ for $i< j$ Equation(2.5) $$=(\frac{(N-n)}{n(N-1)})\sum_{i=1}^N \sum_{j=1}^N (Y_i-Y_j)^2 $$ for $i< j$ ...
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13 views

Help simplifying this sum $f(x) =\sum_{n=1}^{\infty} \frac{2x}{n} e^{-x^2/n} 2^{-n}$, $ x \ge 0$

I am stuck on this sum $f(x) = \sum_{n=1}^{\infty} \frac{2x}{n} e^{-x^2/n} 2^{-n}$ $ x \ge 0$ Any tips on how to get started? Thanks for any help
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1answer
37 views

Are the following Stopping Times?

I've been working through the following list of stopping time questions. I am have problems with the final two (e and f). I appreciate any assistance offered. $\textbf{Question:}$ Let $S,T : ...
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3answers
329 views

How do i visualize Cosets of a group

The Lemma asserted in Herstein as given by $[a] = Ha$ seems very non intuitive to me. How do I think in order that this thing makes sense to me? LEMMA 2.4.4 For all $a$ in $G$ , $$Ha = \{ x \in ...
3
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1answer
85 views

What is the prerequisite knowledge for Navier–Stokes Existence and Smoothness problem?

I am highly interested in the Millennium Problem of Navier–Stokes Existence and Smoothness (also here) and my aim is to reach some level of knowledge to do research on it. The problem seems simple to ...
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4answers
55 views

Prove $R$ is an equivalence relation.

I think I'm on the right track. Set $S = N \times N$, and for any two members $(a,b),(c,d)$ of $S$, define $(a,b) \simeq (c,d)$ provided that $ad = bc$. Prove that $\simeq$ is an equivalence ...
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1answer
31 views

Determinant of an almost-diagonal matrix

I would like to compute the determinant of the $(k+1)\times (k+1)$ matrix below $$J=\begin{vmatrix} y_{k+1}& 0 & \ldots & 0 & y_1 \\ 0& y_{k+1}& \ldots& 0& y_2 \\ ...
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39 views

Suppose a finite set G is closed under associative product and that both cancellation laws hold in G.Prove G must be a group

Suppose a finite set G is closed under associative product and that both cancellation laws hold in G.Prove G must be a group I somehow need to prove identity and inverse ,closure holds to prove that ...