Questions about the process of studying mathematics without formal instruction.

learn more… | top users | synonyms (1)

-1
votes
0answers
16 views

Expectation of a function of random varaible

This may seem a trivial Question but I am confused and never come across this kind of expression. I have an expression $E\bigg [\frac{{(\log(R^p)})^2}{N} \bigg]$ where N = number of data points and ...
2
votes
0answers
16 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, ...
0
votes
0answers
36 views

Unable to solve the integration and derivative of log-likelihood expression

There is an expression which has an integral: $L_x = \ln[nf(x) + \ln V(m) + \ln m]N_{t_K} + (m-1) \int_0^{t_K} \ln (t) dN_t - nf(x)V(m) \int_0^{t_k}mt^{m-1} dt$ $ = \ln[nf(x) + \ln V(m) + \ln ...
1
vote
0answers
10 views

Showing there is a cartesian coordinate system on EG.

I'm pretty sure I should just show there is a bijection between the points in EG and elements of R^2. How do I do this? note: EG=Euclidean Geometry
1
vote
1answer
18 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 $X_1-\Sigma_{12} \Sigma_{22}^{-1}X_2$ ...
0
votes
0answers
37 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 ...
0
votes
1answer
37 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 ...
0
votes
0answers
9 views

On Conditional distribution of the multivariate normal.

Following the answer to this question. Where we are talking about a multivariate normal than has mean and covariance matrix that can be decomposed as: $\boldsymbol\mu = \begin{bmatrix} ...
4
votes
3answers
67 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 ...
-1
votes
1answer
20 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 ...
1
vote
0answers
8 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 ...
1
vote
1answer
21 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 ...
3
votes
2answers
39 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 ...
0
votes
0answers
15 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 ...
1
vote
0answers
23 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 ...
1
vote
0answers
17 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 ...
1
vote
1answer
40 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 ...
3
votes
1answer
44 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
votes
0answers
9 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
vote
1answer
26 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
53 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 = ...
8
votes
2answers
142 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 ...
2
votes
2answers
304 views

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

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
votes
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
vote
2answers
54 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
votes
2answers
63 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 ...
1
vote
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
50 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
votes
3answers
283 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
vote
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
votes
7answers
632 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
43 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
votes
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
50 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
votes
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
votes
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
votes
2answers
49 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
vote
4answers
70 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
votes
1answer
128 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
votes
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
votes
0answers
70 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
votes
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
votes
1answer
87 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
votes
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
votes
2answers
59 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
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 = ...