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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3answers
183 views

Brief book on calculus to read before studying the analysis [closed]

S.E advisers, I am going to start studying the analysis texts (Rudin-PMA, Apostol-MA, Pugh-RMA) on the first week of August. I have a good proof skills through working on Artin's Algebra and ...
1
vote
1answer
41 views

Expectation and Poisson process

Let {$N(t),t\geq0$} be a Poisson process with rate $\lambda$. Calculate $E[N(t).N(t+s)]$ I know that $N(t)\sim Poisson(\lambda t)$ and $N(t+s)\sim Poisson(\lambda(t+s))$ I can assume that ...
7
votes
1answer
209 views

Reading mathematics at the graduate level - does every single proof matter?

I would be interested in hearing from PhD students (and upwards) specializing in pure maths (in particular, the more algebraic aspects). My question is this: When reading to learn mathematics at ...
2
votes
2answers
39 views

Poisson Process proof that

For a Poisson process show, for $s<t$ that $$P(N(s)=k\mid N(t)=n)={n\choose k}\left(\frac{s}{t}\right)^k\left(1-\frac{s}{t}\right)^{n-k},\space > k=0,1,\dots,n$$ I tried a few things but ...
2
votes
1answer
27 views

Notation - matrix calculatoin

i am self-studying linear algebra, and came across the following statement about singular value decomposition (X = $USV^T$). I am somewhat confused on how to interpret the (|) and (--) symbols. Should ...
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0answers
37 views

Derivation - EM algorithm

I am self-studying the EM algorithm from Kevin Murphy's book (http://www.cs.ubc.ca/~murphyk/MLbook/index.html), and have a question that i am struggling with: Moving from step 11.24 and 11.25 - why ...
0
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1answer
104 views

Equivalent conditions of Lebesgue measurable sets

Hi I'd appreciate if someone can check the following exercise any suggestions are welcome. Thanks ;) Let $A$ a subset of ${\bf{R}}^d$ show that the following conditions are equivalent: (i) $A$ ...
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0answers
22 views

Singular value decomposition - unique determination

i am self-studying SVD - and stumbled upon the Wikipedia page (https://en.wikipedia.org/wiki/Singular_value_decomposition) on the statement that a common convention is to order the singular values in ...
0
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0answers
18 views

low-rank matrix decomposition

i am self-studying factor analysis - and the text states that the covariance matrix is approximated with a low-rank matrix decomposition. i am trying to understand what this means in practice - and ...
1
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2answers
102 views

Expectation and waiting time

There are three jobs that need to be processed, with the processing time of job $i$ being exponential with rate $\mu_i$. There are two processors available, so processing on two of the jobs can ...
2
votes
1answer
99 views

Failure time and exponential distribution

One hundred items are simultaneously put on a life test. Suppose the lifetimes of the individual items are independent exponential random variables with mean $200$ hours. The test will end when ...
0
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1answer
42 views

Bounded Matrix-Vector Multiplication

Given a $p\times p$ square matrix $A$. Can I say that the 2 norm of their product is always bounded for any $p \times 1$ vector, please? That is, $$ \| Ax \| <\infty, \forall x\in\mathbb R^p. $$ ...
2
votes
1answer
74 views

Expectation with exponential random variable

If $X_i$, $i=1,2,3$ are independent exponential random variable with rates $\lambda_i$, find $$E[\max(X_i) \mid X_1<X_2<X_3]$$ I really did not understand this exercise, because if ...
3
votes
1answer
155 views

What are some easy to understand applications of Banach Contraction Principle?

I know that Banach contraction principle guarantees a unique solution to problems of the form $$f(x) = x$$ But for the life of me I cannot understand why this problem is important at all. I don't ...
3
votes
0answers
172 views

Inquiry about My Self-Study Plan for Real Analysis (associated with my undergraduate research) [closed]

S.E advisers, I am a college sophomore in US with a major in mathematics and an aspiring mathematician in the computation theory and cryptography. I recently got an undergraduate research in the ...
1
vote
1answer
62 views

Probability with exponential random variable

Machine $1$ is currently working. Machine $2$ will be put in use at time $t$ from now. If the lifetime of machine $i$ is exponential with rate $\lambda_i=1,2$, what is the probability that ...
2
votes
1answer
186 views

log normalizer - exponential family

i am studying the exponential family- and read that, for $p(x|\mu)=h(x)exp(\eta^T t(x)-a(\eta))$, that $a(\eta)$ is the log normalizer, which ensures that the probability distribution integrates to ...
1
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0answers
18 views

Characterize the $\mu^*$- measurable sets where $\mu^∗ = \lambda^* \circ \text{proj}_1 $ and $ \lambda^*$ is the Lebesgue outer measure

Hi I'm working with Cohn's book and I have other problem with the necessity condition, I'd appreciate any help. Let $\lambda^*$ the Lebesgue outer measure on $\bf{R}$, and let $\pi$ be the ...
0
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2answers
51 views

Splitting the summation sign

i am trying to understand the second step in the formula per below - and how the summation sign $\sum_{k=1}^K$ splits into the terms 1-$\sum_{k=1}^{K-1}$ terms. Any help much appreciated
1
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3answers
233 views

Exponential distribution and poisson process

Consider a post office with two clerks. Three people, A, B, and C, enter simultaneously. A and B go directly to the clerks, and C waits until either A or B leaves before he begins service. What ...
4
votes
1answer
71 views

Subsets $B$ of bounded subinterval $I$ is lebesgue measurable iff $\lambda^*(I)=\lambda^*(B)+\lambda^*(I\cap B^c)$

Hi I was reading Cohn's book and I have problem with the following exercises (only the return of b is what I don't know), I'd appreciate any help and suggestion, if necessary, for a): a) Show ...
3
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0answers
50 views

Looking for Math books recommendations to study Electronics

My background is the very basics, and I mean, literally, I can add, sub,mul,div and a little of algebra (near, nothing) and that's it. As you can see I need the best Total Beginner Book(s) that can ...
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1answer
44 views

What is the importance that an assumption needs to state whether a space is Banach space?

I am self studying functional analysis and I don't not see the utility of authors trying make it clear that a space $X$ is a Banach space before proceeding with a definition. For example, going ...
2
votes
1answer
29 views

Outer measure exclusion of zero set

I've just started self-studying measure theory by reading Pugh's Mathematical Analysis. He shows that the exclusion of a zero set does not change the outer measure: $m^*(E\setminus Z)=m^*(E)$, but ...
2
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2answers
55 views

Matrix Differentiation

Consider a differentiable function $f: \mathbb R \to \mathbb R$ and two $p\times 1$ vectors $x$ and $\theta$. Then define a new function as follows. $$ f\left( x^T\theta \right)x. $$ Now we want to ...
3
votes
1answer
105 views

Doubly stochastic matrix proof

A transition matrix $P$ is said to be doubly stochastic if the sum over each column equals one, that is $\sum_i P_{ij}=1\space\forall i$. If such a chain is irreducible and aperiodic and ...
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2answers
41 views

Proof that state can be reached

Prove that if the number of states in a Markov Chain is $M$, and if state $j$ can be reached from state $i$, then it can be reached in $M$ steps or less. To me it just seems the definition of ...
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1answer
70 views

If $R$ is a simple Artinian ring, then when is a finitely generated module free?

Here's an exercise from my book, which only gives a brief solution which leaves me very confused. Let $R$ be a simple Artinian ring, say $R=K_r$. Show that there is only one simple right ...
2
votes
2answers
58 views

Notation - “' sign” in summation

I am studying logistic regression - and i saw the following symbol: c' (in sum symbol of the denominator). What does this mean? I thought it might be all other elements of the vector except the one ...
2
votes
0answers
59 views

equivalent form of almost sure convergence

Consider random variables $X_1, X_2, \dots$ and $X$ on $(\Omega, \mathcal F, \mathbb P)$. We say that $X_n$ converges to $X$ almost surely if $$\mathbb P\left(\lim_{n \to \infty} X_n =X\right)=1.$$ It ...
0
votes
1answer
39 views

Convexity proven as max of linear functions

i am studying convexity, and stumbled upon the statement and example below. Am i right to understand that the function in the example is convex because maximizing the equation on the right hand size ...
2
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1answer
59 views

Introduction to Lebesgue Integration for Statistical Use

I am studying statistics at the graduate level and have a moderate background in real analysis however I unfortunately have no experience with Lebesgue integration. Does anyone have some recommended ...
0
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1answer
54 views

Seeking the Recommendation on Complexity Theory books

S.E advisers, I am a rising college junior in US with a major in mathematics and an aspiring applied mathematician in the fields of theoretical computing. I just recently got a research project on ...
2
votes
1answer
163 views

If $P^r$ has all positive entries, then so does $P^n$

Let $P$ be the transition probability matrix of a Markov Chain. Argue that it for some positive integer r, $P^r$ has all positive entries, then so does $P^n$, for all integers $n\geq r$ I ...
0
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1answer
41 views

Vector spaces - $\mathbb{R}^n$ and $\mathbb{R}^m$

I stumbled on the following text on Wikipedia: Suppose the random column vectors X, Y live in $\mathbb{R}^n$ and $\mathbb{R}^m$ respectively, and the vector $(X, Y)$ in $\mathbb{R}^{n+m}$ has a ...
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0answers
65 views

Understanding the formula

Let $P$ the transition probability matrix and $\mu$ the row vector of initial distribution. $$P_\mu(X_n=j)=\sum_j\mu(i)p^n(i,j)=\mu p^n(j)$$ I don't want to make a proof of that, I want to ...
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1answer
34 views

Expectation in markov chain

A Markov Chain {$X_n,n\geq0$} with states $0,1,2$, has the transition probability matrix ...
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0answers
28 views

Looking for resources on these topics from Linear Algebra

I am looking for Characteristic roots and characteristic vectors of a linear transformation or of a matrix, Algebraic and Geometric multiplicity of a characteristic value, Cayley-Hamilton theorem, ...
0
votes
1answer
158 views

Is {$X_n,n\geq 0$} a markov chain?

Consider a process {$X_n,n=0,1,\dots$}, which takes on the values $0,1,2$. Suppose $$P(X_{n+1}=j|X_n=i,X_{n-1}=i_{n-1},\dots,X_0=i_0)$$ $$=P_{ij}^I,\text{when n is even}$$ ...
0
votes
1answer
25 views

Is the function $G$ right continuous?

Suppose $S\subset\mathbb R$ such that $S\cap(x,\infty)\neq\phi$ for every $x\in\mathbb R$ and suppose that $g:S\to\mathbb R$ is a non-decreasing bounded function. Define $G:\mathbb R\to\mathbb R$ ...
1
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2answers
107 views

Question from self-studying Halmos' Finite Dimensional Vector Spaces

For section 1 on Fields, there is a question 2c: 2. a) Is the set of all positive integers a field? b) What about the set of all integers? c) Can the answers to both these question be changed by ...
0
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1answer
62 views

To show a function is Riemann integrable.

Given a function $$v(x) =\begin{cases} 0&\text{if } x=0\\ 1&\text{if }x\in(0,1]\end{cases}$$ How do I show that $v$ is Riemann integrable in $[0,1]$? Hints on how to do this??
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2answers
187 views

Which should I study first: Logic or set theory?

I'm an undergraduate student in a college of sciences and technics studying maths, physics, computing and some chimestry so we studied elementary materials in logic and set theory. As I am interested ...
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0answers
43 views

To prove $| S(f,P,T) - S(g,P,T) | \leq M(b-a)$ ( Riemann Integration)

To prove $| S(f,P,T) - S(g,P,T) | \leq M(b-a)$ Question : Let $[a,b] \subseteq R$ be a non degenerative closed bounded interval and let $f,g :[a,b] \rightarrow R$ be functions .Suppose that there is ...
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0answers
11 views

A reference to study Boundary conditions of diffusion processes

I am trying to learn about Wentzell Boundary condition and (A,L) diffusion in the sense of Watanabe's paper (On the existence and uniqueness of diffusion processes with Wentzell's boundary condition ...
2
votes
2answers
178 views

What is the best way to study Probability? [closed]

Nowadays, I am studying probability. I want to be an actuary and the first exam that I have to pass is P exam. I just want to know what is the best way and if you can recommend any books please let me ...
2
votes
3answers
87 views

What does the quotient group $(A+B)/B$ actually mean?

I understand that $A+B$ is the set containing all elements of the form $a+b$, wit $a\in A, b\in B$. When you do the quotient group, that's like forming equivalence classes modulo $B$. All elements ...
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2answers
45 views

Proof the statement

Given a finite aperiodic irreducible Markov Chain, prove that for some $n$ all terms of $P^n$ are positive. I'm little lost in how to prove that, but I know that: $i)$ If a Markov Chain is ...
3
votes
1answer
45 views

Stochastic matrix proof

Every stochastic $n\times n$ matrix corresponds to a Markov chain for which it is the one-step transition matrix. However, not every stochastic matrix $n\times n$ is the two-step transition ...
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1answer
43 views

Periodicity of states in Markov Chain

Determine the classes and the periodicity of the various states for a Markov Chain with transition probability matrix ...