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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An Example of Taylor Series

Consider a positive function $f(x)$ and suppose that we would like to approximate its value around some point $x_0$. One way to do so is to use two-term Taylor series expansion as follows. $$ f(x) \...
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1answer
78 views

What are the steps to this functional derivative problem?

I'm trying to derive equations from Matthew Beal's Thesis, Variational Algorithms for Approximate Bayesian Inference pg.55, but I'm stuck on one of the equations (well I'm stuck on a lot of equations ...
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1answer
51 views

Find the extremals of a functional of the form $\int^{x_1}_{x_0}F(y',z')dx$

I was working on Problem 3 in Ch. 2 of Gelfand & Fomin's Calculus of Variations, which reads: Find the extremals of a functional of the form $$\int^{x_1}_{x_0}F(y',z')dx$$ given that $F_{y'y'}F_{...
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63 views

Proving Frobenius Theorem for Eigen Values

In my mulitivariable calculus class to justify second derivative test my professor used a theorem he called the frobenius theorem. But when I searched on wiki all I could find was Perron Frobenius ...
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57 views

Calculating Power of a Paired T Test

$ 239$ subjects had their cholesterol measured, and then were put on high-fiber diets. After a month on the high-fiber diet, the cholesterol was measured again. The mean LDL cholesterol level before ...
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2answers
99 views

Strange behaviors of finitely additive probabilities

Watching a lecture on youtube I heard the lecturer stating that in general finitely additive probabilities behaves strangely. For example, it is possible that every open interval around a point $x$ ...
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1answer
28 views

How to determine a function whose minima falls on a specified curve?

I have a family of curves given by $g(x,y)=C_0 yx^{-n}$. How can I determine the function $f(x,y)$ for the family of curves that satisfies the condition that the local minima $\frac{\partial f}{\...
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1answer
75 views

Learning from Alternative Sources

I have a very general question about people's experiences with learning math. I can think of a couple of times where I had the following situation. I was seeking to learning about topic A. However, ...
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20 views

Showing there is a cartesian coordinate system in Euclidean geometry.

I'm pretty sure I should just show there is a bijection between the points in Euclidean Geometry and elements of $\mathbb{R^2}$. How do I do this?
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1answer
24 views

Independence of two multivariate normals.

Suppose we have two multivariate normals $X_1 \sim N(u_1, \Sigma_{11}\Sigma_{22}$) and $X_2 \sim N(u_2, \Sigma_{21} \Sigma_{22})$ . Why are $X_2 $ and $X_1-\Sigma_{12} \Sigma_{22}^{-1}X_2$ ...
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81 views

What are some interesting, atypical mathematical topics that a student who has taken an introductory calculus sequence can learn about?

I understand that usually the next step after $3$ semesters of calculus and $1$ semester of ordinary differential equations (plus one semester of linear algebra, for some) is something like an ...
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57 views

Conditional expectation of $X$ given $Z$, where $Z = 1$ if $X > Y$ and $-1$, otherwise

Let $X\sim\operatorname{Exp}(1)$ and $Y\sim\operatorname{Exp}(2)$ be independent random variables. Define $Z$ by $$ Z = \begin{cases} 1,& X>Y\\ -1,& X\leqslant Y. \end{cases} $$ I want to ...
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4answers
669 views

Outline for high school combinatorics class?

I am a high school student and I have taken all the math classes that my school provides (through calculus AB). I have been looking at a possible independent study for next year and I have landed on ...
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1answer
260 views

sum of two dependent random variables

Let $X$ be a cotinuous random variable uniformly distributed over $[-10,10]$. Let $Y$ be a random variable with pdf $f_Y(y) = \frac{1}{40}\ln \frac{20}{|y|}, -20 \leq y \leq 20$. $X$ and $Y$ ARE NOT ...
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0answers
21 views

transformation and functions of random variables

Let $X,Y$ be independent random variables. I already have the distribution of $XY$ over a certain subinterval of $\mathbb{R}$, by convolution. My question is, is it possible to get the distribution of ...
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1answer
60 views

The inverse of the sum of two matrices in *Applied statistical decision theory *.

I am following Applied statistical decision theory [by] Raiffa, Howard. Which can be consulted online here. A theorem at the page linked states that if two matrices $A,B$ are non-singular and of ...
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2answers
69 views

Logic behind a proof in Topological Vector Spaces

I found the following result at the beginning of some notes on topological vector spaces (TVS). This is a quite fundamental result, that apparently is considered the corresponding version of the ...
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1answer
54 views

A math proof within a question about homogeneous Poisson process

We know that a homogeneous Poisson process is a process with a constant intensity $\lambda$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right \...
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0answers
72 views

sum/product combination of random variables

Let $X$ and $Y$ be independent random variables. If I am asked about the distribution of random variable $XY+Y$, is it ok if I compute $XY$ first and then add the result to $Y$ (via convolution, or ...
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0answers
41 views

Proof about a homogeneous Poisson process

We know that a homogeneous Poisson process is a process with a constant intensity $\lambda$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right \...
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1answer
72 views

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

In the book "Lehrbuch der Analysis Teil I" of Heuser page 303, there was a task: Prove $$\frac{d}{dx}{\rm arctanh}(\ln \cosh x) = \frac{\tanh x}{1-(\ln \cosh x)^2}.$$ When I tried, I ended up with $$\...
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1answer
73 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: $$F_z(z+H(z))-F_z(z-H(z))=0....
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1answer
34 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 ...
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1answer
125 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 = 1(x_{...
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2answers
547 views

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 ...
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2answers
305 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 $p(...
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2answers
439 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 ...
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7answers
265 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 ...
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1answer
37 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 ...
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3answers
318 views

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

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 ...
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1answer
63 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 ...
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7answers
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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 ...
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1answer
461 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 ...
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1answer
887 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 ...
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1answer
29 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, ...
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2answers
91 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 start ...
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4answers
79 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. ...
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260 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 ...
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79 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: $$(...
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0answers
97 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 ...
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0answers
84 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 ...
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1answer
83 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 ...
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1answer
47 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 ...
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1answer
362 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 ...
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13answers
5k 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 ...
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2answers
67 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 ...
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1answer
37 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 = ...
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1answer
31 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 ...
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1answer
49 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 ...
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0answers
41 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 ...