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_.

learn more… | top users | synonyms (1)

2
votes
0answers
50 views

Proof of Heisenberg Uncertainty Principle Exercise

I'm not very knowledgeable in QM, and I know many physics books derive the uncertainty principle using commutators, but as an exercise in my PDE book (by Asmar), I should be able to derive it from one ...
0
votes
1answer
60 views

Which topics in maths should I know before I dive into programming for image processing?

I am a student who wants to start out with programming for Image processing but as I do not have a good mathematical background(I haven't studied A-level Maths) I would like to know what are the ...
1
vote
1answer
34 views

Continuity Set of Monotone Functions

Let $f$ be a real-valued monotone function defined on an interval $I$. Then we know that the set $D \subset I$ of discontinuities of the first kind is at most countable. Then can I say that the ...
0
votes
0answers
16 views

Convergence of Distribution Functions

This is paragraph from de Haan's Extreme Value Theory (2006, p4). Let $F$ be a cumulative distribution function, $a_n$ a sequence of positive constants and $b_n$ a sequence of real numbers. Suppose ...
7
votes
1answer
197 views

“Visualizing” Mathematical Objects - Tips & Tricks

It has been a while since I am kind of stuck with my skills concerning the visualization of mathematical objects. Here there is the problem. First of all, let me point out that I am completely ...
52
votes
13answers
4k views

How to stop forgetting proofs - for a first course in Real Analysis?

I am taking my first course in analysis. I like the subject. I study it almost on a daily basis. I try to prove theorems on my own without even looking at the hints. If I really get stuck I just read ...
0
votes
2answers
61 views

Proving that if $\sum_{k = m}^{\infty}P(A_k) < \infty$ then $\lim_{m \rightarrow \infty}\sum_{k = m}^{\infty}P(A_k) = 0$.

I want to prove that if $\sum_{k = m}^{\infty}P(A_k) < \infty$ then $\lim_{m \rightarrow \infty}\sum_{k = m}^\infty P(A_k) = 0$. Bu I am not quite there, I will write where I got to trying to do ...
0
votes
1answer
31 views

Proving $P \bigg( \bigcup_n \bigcap_{k = n}^{\infty}A_k \bigg) = lim_{n \rightarrow \infty}P \bigg( \bigcap_{k = n}^{\infty}A_k \bigg) $?

$P$ is a probability measure and $A_1, A_2, ... \in F$ that is a sigma algebra. $$P \bigg( \bigcup_{n=1}^{\infty} \bigcap_{k = n}^{\infty}A_k \bigg) = lim_{n \rightarrow \infty}P \bigg( \bigcap_{k = ...
1
vote
1answer
29 views

Confidence interval - determining the Confidence based on pre set upper and lower boundaries.

I am trying to solve home made problem, but i am having a hard time solving it.. A Appleseller wants to advertise the average weight of his apple, but since he sells so many it isn't possible to do ...
0
votes
1answer
39 views

On a limit of random variables.

This is a duplicate of this question that has not got an answer. I am going to try to improve my question that is probably missworded since I do not believe it to be difficult, even though I can't ...
1
vote
0answers
27 views

What does the Gamma means in local ringed space?

I found the following problem from an algebraic geometry course hold in 2003. Let $(X,\mathscr A)$ locally ringed space and $f\in \Gamma(X,\mathscr A)$. Prove that $$X_f=\{x\in X|f(x)\ne 0\}$$ is an ...
3
votes
2answers
106 views

Do we have $-1\bmod 2 \equiv -1$ or $+1$?

As far I can calculate $-1 \bmod 2 \equiv -1$, but the software I am using (R) is telling that $-1 \bmod 2 \equiv +1$. This is the R code: -1%%2 [1]1 Which is ...
1
vote
2answers
40 views

Find the expected value of $Z=2Y_1+Y_2$ where $Y_1=\min(X_1,X_2),Y_2=\max(X_1,X_2)$ and $X_i$ is exponential with $\theta=2$

Where each $X_i$ is independent. I know $E(X_i)=2$. So: $$E(Z) = E(2Y_1 + Y_2) = E[2\min(X_1,X_2) + \max(X_1,X_2)] = E\{2(X_1 or X_2) + (X_1 or X_2)\}$$ Since regardless of the outcome of the min and ...
0
votes
0answers
40 views

Optimization 101 for electrical engineers…Where to start from?

I have never taken any optimization class. From an electrical engineering point of view, how should I approach learning this field? What kind of information I should be looking at in my problem to ...
1
vote
1answer
35 views

Proving this set is an algebra.

Let $J = \{$all intervals contained in $[0,1]\}$ and $B_0 = \{$all finite unions of elements of $J\}$. Prove that $B_o$ contains $[0,1], \emptyset$, and is closed under formation of complements and ...
0
votes
1answer
38 views

Given $Var(X)=8100,Var(Y)=10000,Var(X+Y)=20000$, calculate $Var(X+500+(1.08)Y)$.

In considering medical insurance for a certain operation, let $X$ equal the amount (in dollars) paid for the doctor and let $Y$ equal the amount paid to the hospital. In the past, the variances ...
4
votes
1answer
90 views

deduce that $\cos 6° \cos42° \cos66° \cos78°= \frac{1}{16}$

Prove that $$4 \cos\theta \cos(\frac{\pi}{3}-\theta) \cos(\frac{\pi}{3}+\theta)= \cos 3\theta$$ and deduce that $$\cos 6° \cos42° \cos66° \cos78°= \frac{1}{16}$$ I have proved by using $2 \cos A ...
1
vote
2answers
64 views

What is $P(X<Y)$ when $X,Y$ represent a random sample from two distributions.

The income of people in two cities is represented by two Pareto-type pdfs: $$f(x)=\frac{2}{x^3},1<x<\infty,g(y)=\frac{3}{y^4},1<y<\infty$$ One person is selected at random from each ...
0
votes
2answers
71 views

(Geometric algebra) Acceleration of a particle with constant speed as a bivector-vector inner product

I've been working on (self-studying) Geometric Algebra for Physicists which, sadly, has no solutions manual. This is not a problem in general, but I feel like one of my solutions for a question asked ...
1
vote
1answer
26 views

Find the mean and the variance of an F random variable with $r_1$ and $r_2$ degrees of freedom.

First find $E(U), E(\frac{1}{V}), E(U^2),E(\frac{1}{V^2})$. When I consider finding $E(U)$ I feel as though integrating over the pdf of the F distribution multiplied by $u$ will leave me with a ...
0
votes
1answer
46 views

Determine the lifetime of a device with two components that follow exponential distributions

Let $X_1,X_2$ be independent random variables representing the lifetimes (in hours) of two key components of a device that fails when and only when both components fail. Say each $X_i$ has an ...
5
votes
6answers
203 views

Is integration by parts the best method for $\int_0^1 x^3(1-x)^6 dx$?

It came up when finding a constant such that the integral is equal to 1 and thus behaves like a pdf. I used the parts method but have made an error, just curious how others might approach the problem. ...
2
votes
2answers
73 views

Let $Y = X^2$. Find the pdf of Y when the distribution is $N(0,1)$. [duplicate]

I've performed a change of variable: $$X = \sqrt{y}$$ $$X'=\frac{1}{2}Y^{-\frac{1}{2}}$$ Thus: $$f(\sqrt{y})*X'=f(y)=\frac{1}{2\sqrt{2\pi y}}e^{-\frac{y}{2}}$$ However the book gives: ...
0
votes
1answer
39 views

On why a set is measurable ( a step of the proof of the Lebesgue monotone convergence theorem in Rudin).

Let $\{f_n\}$ be a sequence of measurable functions on $X$, and suppose that, for every $x \in X$: $$0 \le f_1(x) \le f_2(x) \le \cdots \le \infty $$ and $f_n(x) \rightarrow f(x) $ as $n \rightarrow ...
5
votes
4answers
144 views

Reading material for highschool mathematics

my name is gaurav. I'm presently self-employed aged 32. During my school days i was unable to study maths & physics in a manner that cleared my basic concepts. When i got hold of the way,it was ...
0
votes
2answers
42 views

Frequency of sinusoidal curve

In this site,The frequency of a trigonometric function is defined as the number of cycles it completes in a given interval. The formula is : frequency=1/period The period of a sine function is ...
0
votes
1answer
49 views

If $f : X \rightarrow [ -\infty, +\infty]$ is a measurable function is $\{x : f(x) < \alpha \}$ a measurable set?

If $f : X \rightarrow [ -\infty, +\infty]$ is a measurable function is $\{x : f(x) < \alpha \}$ a measurable set? Where $\alpha \in \bar{R}$. It seems to me that it entirely depends on the range ...
1
vote
5answers
99 views

Solve $\sin(x)=-\frac{1}{2}$

I have to solve the following equation for $x$ $$\cos(2x)+\sin(x)=0$$ After simplification i got $\sin(x)=1$ or $\sin(x)=-\frac{1}{2}$ $\Rightarrow x=90^0$ But don't know how to solve for $x$ ...
0
votes
1answer
86 views

Which topology textbook has the greatest amount of ancillary support available on the Internet?

I'm considering to begin upon the study of topology and am wondering which book would the best option. I've even started reading Munkres and G. F. Simmons, but the problem is neither book has any ...
1
vote
1answer
45 views

Describe the Calculus Identity $f(x) = f(a-x)$

To me this reads: "A constant minus x is equivalent to x on $[-\infty, \infty]$".
2
votes
2answers
53 views

What is the definition of closed subspace?

I am trying to understand what is intended with closed subspace, I took the following guess: A closed subspace $M$ of a Hilbert space $H$ is a subspace of $H$ s.t. any sequence $\{x_n\}$ of elements ...
3
votes
3answers
62 views

On the intuition behind the projection theorem.

I have recently proved the projection theorem in a Hilbert space setting. The statements were: If $M$ is a closed subspace of a Hilbert space $H$ and $x \in H$, then: There is a unique element ...
6
votes
2answers
226 views

For someone who is self-studying topology: what are the main topics to focus on?

I will have to teach myself topology for the Math GRE Subject Test because, although I graduated with a math major, I never took topology. I have Munkres and Kelley, along with the Schaum's Outlines ...
7
votes
3answers
95 views

What does one study to increase understanding of the $P \stackrel{?}{=} NP$ problem?

If one were to learn more about the $P \stackrel{?}{=} NP$ problem, where would one start? I understand what the problem is—but not enough to be able to read anything technical about it. ...
2
votes
1answer
61 views

Length of A Diagonal Line of Square

After watching this video to calculate the length of diagonal of square , a question arises to me is : Why is length of diagonal of square $$\frac{\text{side}}{cos45^0}$$? Why $cos45^0$ ?If i ...
8
votes
2answers
208 views

Importance of the zero free region of Riemann zeta function

I have heard that for improving the error term in the Prime Number Theorem, we need better and better estimates on the zero free region. I have also heard that the best possible error term comes from ...
1
vote
1answer
52 views

Let $X$ and $Y$ have the joint pdf $f(x,y)=8x(1-y),0<y<1,0<x<1-y$. Compute $P(Y<X\mid X \leqq \frac{1}{4})$

Let $X$ and $Y$ have the joint pdf $f(x,y)=8x(1-y)$, $0<y<1$, $0<x<1-y$. Compute $P(Y<X\mid X \leqq \frac{1}{4})$. I know: $$P(A\mid B) = \frac{P(A \cap B)}{P(B)}$$ Therefore I ...
1
vote
2answers
51 views

What is the probability that $X$ and $Y$ are within 0.1 of each other given a uniformly distributed joint pdf.

Two construction companies make bids of $X$ and $Y$ (in \$$100,000$'s) on a remodeling project. The joint pdf of $X$ and $Y$ is uniform on the space $2<x<2.5,2<y<2.3$. If $X$ and $Y$ ...
0
votes
0answers
24 views

On why a limit of random variables implies $Z_k(w)\leq z+\frac{1}{m}$.

If $Z_1,Z_2,\ldots$ are random variables such that $\lim_{n \to \infty}Z_n(w)$ exist for all $w$ and $$Z(w)=\lim_{n \to \infty}Z_n(w)$$ and suppose $w\in\{Z^{-1 }\mid Z\leq z\}$, for $z \in \mathbb ...
1
vote
1answer
79 views

Probability and Measure Theory by Ash

Has anyone used the textbook above? If so how does it compare with billingsley, Chung and similar such books in terms of rigor, coverage, and ease if use for self study?
0
votes
0answers
46 views

Find the marginal pdf of a joint distribution

Let $X$ and $Y$ have a uniform distribution on the set of points with integer coordinates in $S = \{(x,y):0\leqq x\leqq7, x \leqq y \leqq x+2\}$. That is, $f(x,y) = \frac{1}{24}\in S$, and both x and ...
2
votes
1answer
38 views

How to find all Dirichlet characters

I want to know all the Dirichlet characters modulo $m$. I know that the number of such characters are $\phi(m)$. But how do find each and every character. for small moduli I could do it using some ...
0
votes
0answers
35 views

Calculus of Variations-First and Second Order Deviations

I'm new to Calculus of Variations and the Method of Least Action (L=T-V) What I'm unsure about is how first and second order deviations are used in finding the least action. I know it's used to find ...
0
votes
1answer
17 views

On the polar representation of an inner product.

Take $H$ an inner product space. $x,y \in H$. Take $b = |<x,y>|$ . Then the polar representation of $<x,y>$ is: $$<x,y> = be^{i\theta}$$ for some $\theta \in (-\pi, \pi]$. Why is ...
0
votes
2answers
38 views

Difference among the same distribution , identical distribution and similar distribution.

$X\sim N(\mu_1,\sigma)$ and $Y\sim N(\mu_2,\sigma)$ are similar but not identical. $X\sim N(\mu,\sigma)$ and $Y\sim N(\mu,\sigma)$ are identical. But what is same distribution? Do same and ...
1
vote
3answers
131 views

Is natural numbers set $\mathbb N$ infinite set?

A set with uncountable number of elements is called an infinite set. Is that the set of all natural numbers, $\Bbb N=\text{{$1,2,3,\ldots$}}$ infinite set? As far i know $\Bbb N$ is "countably" ...
2
votes
1answer
44 views

Proof of the starting part of theorem 1.17 Rudin ( Complex and Reals)

The proof I would like is of the following fact: Put $\delta_n = 2^{-n}$. To each positive integer n and each real number t corresponds a unique integer $ k = k_n(t)$ that satisfies $k \delta_n \le t ...
0
votes
1answer
37 views

$\mathbb E[\bar X_n]=0$

A conditional normal rv sequence, does the mean converges in probability, in this question how can i get $\mathbb E[\bar X_n]=0$? Here is my attempt; $$\mathbb E[\bar ...
0
votes
1answer
53 views

Partial derivative is bounded

Let $f(t,z)$ be a bounded (say by a constant $M$) continuous function on $\mathbb{R}_t \times \mathcal{U}$ where $\mathcal{U}$ is an open neighborhood of $0 \in \mathbb{C}_z$. Moreover, for each fixed ...
41
votes
2answers
703 views

Effective Research Notes

Note-taking for research is vital to your success as a mathematician. As I look back at some of my handwritten notes, I realized how poor they were. I had thought to myself, "What happened?" I was ...