Questions about the process of studying mathematics without formal instruction.

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0
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1answer
27 views

Prove $\frac{d}{dt}{\rm arctanh}(\ln \cosh x) = \frac{\tanh x}{1-(\ln \cosh x)^2}$

In the book "Lehrbuch der Analysis Teil I" of Heuser page 303, there was a task: Prove $$\frac{d}{dt}{\rm arctanh}(\ln \cosh x) = \frac{\tanh x}{1-(\ln \cosh x)^2}.$$ When I tried, I ended up with ...
0
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0answers
15 views

How was the explicit closed form for this implicit function derived?

The problem comes from reading this [0] paper but I think I can express it in a self contained question. Consider the implicit function $H(z)$ defined by the relation: ...
0
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0answers
6 views

Equivalence of the partial least square regresssion's iterative algorithm and its optimization problem

I am reading The Elements of Statistical Learning. This is a page from the partial least square section: The exercise asks to prove the equivalence between Algorithm 3.3 and Eq. (3.64). Here's my ...
1
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1answer
21 views

Where can I find simple integration problems (and other computational exercises) involving special functions?

Working lots of computational exercises in my pre-calculus and calculus classes has given me a great deal of intuition in dealing with elementary functions. Thanks to these years of practice, I can ...
2
votes
1answer
51 views

Is this function Riemann integrable in $[0,1]$?

The function is $f(x) = 1$ for $ 0 \le x \lt 1 $ and $f(x) = 2$ for $x = 1$ I calculate the upper sum $$U(P,f) = \sum_{i=1}^n M_i \Delta x_i = \sum_{i=1}^{n-1} 1\,\Delta x_i + 2 \,\Delta x_n = ...
7
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2answers
116 views
+50

Weak topologies and weak convergence - Looking for feedbacks

I am currently trying to get exactly what the weak and the weak* topologies are, in particular in connection to the concept of weak convergence in measure, however I am not completely sure on what I ...
2
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2answers
285 views

An example of a great explanation or freely accessible article on a math concept [on hold]

Question: Give an example of a great explanation or freely accessible article on a math concept (suitable at the undergraduate or lower level), and explain why you think it is great. Possible ...
0
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0answers
47 views

Revisiting maths through self study

I am a practicing commercial engineer having studied 3 Maths courses during undergraduate college (2004-2008). Now I want to return to my real passion i.e. astrophysics/ quantum mechanics on my own. ...
1
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2answers
50 views

Convexity of mutual information $I(X;Y)$ in conditional $p(y \mid x)$

I'm trying to understand the proof that $I(X;Y)$ is convex in conditional distribution $p(y \mid x)$ - from Elements of Information Theory by Cover & Thomas, theorem 2.7.4. In the proof we fix ...
0
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2answers
62 views

How to brush up on calculus?

It's been years since I took calculus, and while I have a good understanding of the theorems of single variable calculus from my real analysis courses, computationally I am a bit slow. It takes me ...
-2
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0answers
29 views

Self Learning Help [closed]

I'm new here. I was just wondering could anyone recommend any good books that would be A-level equivalent Mathematics for self learning? Something that is definitely for beginner level and easy to ...
1
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7answers
151 views

How to estimate the value of $e$. [closed]

I am currently studying how to estimate $e$. To solve this problem I use these methods discuss below: Method 1: We know that $e^x = 1 + \dfrac{x}{1!} + \dfrac{x}{2!}+ \cdots $ So if we consider a ...
1
vote
1answer
32 views

Convexity of $I(X;Y)$: why $H(Y)$ convex in $p(y)$ $\Rightarrow$ $H(Y)$ convex in $p(x)$

I would like to understand the proof that mutual information $I(X;Y)$ is concave in $p(x)$ - as presented in Elements of Information Theory by Cover & Thomas, theorem 2.7.4. Here's the proof from ...
0
votes
0answers
45 views

Apostol's Calculus Vol II OR Hubburd's multivariable OR Shifrin's multivariable for self study

I'm trying to self study multivariable calculus which I took at university but mostly forgot about it! I'm looking for a textbook that also incorporates linear algebra and gives a coherent view of the ...
4
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3answers
278 views

What things should one know in order to enjoy their undergraduate degree?

From looking at undergraduate mathematics programmes it's quite apparent that mathematics degrees are demanding, one could even say the work load is grueling. However I'm certain that there are ...
1
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1answer
36 views

Find the distribution of sum and product of standard normal random variables

Let $X,Y$ and $Z$ be three independent real valued random variables. All with finite second moment and all with mean $0$ and variance $1$. Define $$ W= \frac{X+YZ}{\sqrt{1+Z^2}} $$ Find the ...
6
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7answers
616 views

A book for abstract algebra with high school level

Any book that I find on abstract algebra is somehow advanced and not OK for self-learning. I am high-school student with high-school math knowledge. Please someone tell me a book can be fine on ...
1
vote
1answer
39 views

Expected value of division

Let $X,Y$ and $Z$ be three indenependent real valued random variables. Al with finite second momennt and all with mean $0$ and variance $1$. Define $$ W= \frac{X+YZ}{\sqrt{1+Z^2}} $$ Show that ...
0
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0answers
49 views

Is my intuition on projectivization correct?

Is my intuition on what a projectivization of an affine curve in $C^2$ is and why it is useful correct? From what I understand given an affine curve $C$ we are trying to find a projective curve ...
0
votes
1answer
45 views

Example of a real-world situation where multivariate analysis is applicable.

I have searched a lot of site to understand the situation where multivariate analysis is applicable. But not got any easily understandable example. Would you please give me a real-world example where ...
0
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0answers
20 views

Application of Multivariate Analysis

The following situation is proven valuable where multivariate analysis can be applied. This example is taken from the book Applied Multivariate Statistical Analysis ...
0
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1answer
26 views

Proving $(I+T)^k$ has positive entries for large k

This is mentioned in these slides. A non-negative square matrix $T$ is called primitive if there is a $k$ such that all the entries of $T^ k$ are positive. It is called irreducible if for any$ i, ...
0
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2answers
47 views

Show that $V=\frac{Z_1}{\sqrt{(Z^2_1 + Z^2_2)/2}}$ has pdf $f(v) = 1 / (\pi \sqrt{2-v^2}),-\sqrt2<v<\sqrt2$

Let $Z_1, Z_2$ have independent standard normal distributions, $N(0,1)$. If the random variable in the numerator did not also appear in the denominator this would be a t distribution. Should ...
1
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4answers
69 views

How to show $I_p(a,b) = \sum_{j=a}^{a+b-1}{a+b-1 \choose j} p^j(1-p)^{a+b-1-j}$

Show that $$I_p(a,b) = \frac{1}{B(a,b)}\int_0^p u^{a-1}(1-u)^{b-1}~du\\= \sum_{j=a}^{a+b-1}{a+b-1 \choose j} p^j(1-p)^{a+b-1-j}$$ when $a,b$ are positive integers. I have no idea how to proceed. ...
8
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1answer
124 views

Problems with the proof that $\ell^p$ is complete

By struggling with the proof that $\ell^p$ is complete, I looked up different proofs by different authors, and I ended up focusing on the one given by Kreyszig in his classic book on functional ...
0
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0answers
21 views

Find the probability that at least one of two light bulb survives for 920 hours.

The length in hours $X$ of lightbulb A is $N(800,14400)$ and $Y$ (lightbulb B) is $N(850,2500)$. Find the probability that at least one of the bulbs lives for at least 920 hours. Would this be: ...
2
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0answers
69 views

Background & Advice for a self-learner of Descriptive Set Theory

A rather straight to the point soft-question: What kind of background should have somebody who wants to study properly descriptive set theory? More specifically, how much analysis should she/he ...
2
votes
0answers
35 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
49 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
29 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
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0answers
14 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 ...
6
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1answer
84 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 ...
48
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13answers
3k 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
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2answers
58 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
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1answer
29 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
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1answer
27 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
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1answer
36 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
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0answers
26 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
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2answers
103 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
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2answers
23 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
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0answers
37 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
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1answer
31 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
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1answer
27 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
86 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 ...
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2answers
48 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
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2answers
59 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
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1answer
21 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
20 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
178 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
61 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: ...