Questions tagged [self-learning]
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.
-2
votes
1answer
59 views
How to split using partial fractions: $\frac{9x^2+48x+18}{(2x+1)(x^2+8x+3)}$?
I'm self-studying from Stroud & Booth's amazing "Engineering Mathematics", but am stuck on a problem from the "Partial Fractions" chapter.
I've been running around in circles trying to solve it, ...
3
votes
0answers
73 views
MacLane Birkhoff $\textit{ Algebra }$ vs Jacobson $\textit{ Basic Algebra I,II }$ vs Lang $\textit{ Algebra }$
I'm searching for an apt textbook(s) on Abstract Algebra - in particular I'm trying to get a review/comparison of the aforementioned books. I would be grateful for any help in this direction.
$$$$
A ...
1
vote
1answer
42 views
Topics needed to learn to understand the math in this paper? GAN - Goodfellow et al.
I am reading this paper by Ian Goodfellow. Currently I have an undergraduate knowledge of probability/statistics/linear algebra/calculus/differential equations. I am trying to understand the formula ...
0
votes
0answers
18 views
Upper bound of floor function
I'm studying a paper where the authors stated preliminarily:
$$\mid R_{i,T}\mid \leq \frac{C \theta_0}{K(i_\theta,\theta,T)}, \forall i\in\{1,\dotsc,\theta^x\}$$
where $C$ is a finite contant; $\...
0
votes
0answers
11 views
Large Deviation, Optimal Transport and Machine Learning Reference
I am looking for references (books/sites/articles) on the following three subjects: Large Deviation, Optimal Transport and Machine Learning References. I would like works which involve any of them ...
0
votes
3answers
51 views
Good way of learning math
In my secondary school, I have been copying the solutions of the questions I could not solve and I guess this helps you to memorize (not blind memorization, sending the info to long term memory). Do ...
0
votes
3answers
45 views
How to evaluate $\sum_{ r=1}^{16}(5r-7)$?
I'm self-studying from Stroud & Booth's "Engineering Mathematics" and in the "Binomials" chapter, one of the last exercises is to evaluate:
$$\sum_{ r=1}^{16}(5r-7)$$
This has got me confused, ...
-2
votes
1answer
27 views
I need to know if.my example for this problem solved
Let $f(x)=\sup(x^3, x^2+1)$, $I=[1,4]$
$\sup(x^3)=64,\ \inf(x^3)=1$
$\sup(x^2+1)=17,\ \inf(x^2+1)=2$
So $\sup(x^3,x^2+1)=\sup(64,17) =64$
Is this correct, can I have good example.
0
votes
0answers
28 views
Where can I study polynomials online?
I have one good book on polynomials. It is called (Level of Knowledge of Polynomials) by its author S. L. Tabachnikov. In principle, I could study it, but I am still mastering the Agebra I. Not long ...
0
votes
3answers
65 views
I am stumped on this problem
It is from Intro to Analysis by Bartle 3rd ed
Let $I:=[a,b]$ let $f:I\rightarrow\mathbb{R}$ be a continuous function
with $f(x)>0$ Prove that there exists a number
$c>0$ such that $f(x)\geq c ...
0
votes
2answers
18 views
Is there a way to write this term different $e^{x+e^x}$?
I am a little confused what this actually means:
$e^{x+e^x}$
It is obviously not the same if I for example
$$e^{x}:= \lambda \\
e^{x+e^x} \neq \lambda^\lambda
$$
0
votes
0answers
35 views
Is the zero truncated Poisson Distribution part of the Exponential Family?
This is the density of a truncated Poisson:
$$P(X = x \mid X > 0) =
\frac{\lambda ^ x e^{- \lambda} }{x ! \left ( 1 - e^{- \lambda} \right )}$$
To show that it's member of the Exponential ...
1
vote
2answers
33 views
Convergence of indicator functions series
Say $E$ is a measurable set, and $\{E_k\}$ is a series of measurable sets defined by
$$
E_k \subset E, m(E \setminus E_k) < \frac{1}{k}, k = 1, 2, 3, ...
$$
Do their corresponding indicator ...
0
votes
0answers
32 views
Puzzles and exercises to improve mathematical intelligence and spatial thinking
In your childhood or adolescence, or maybe as an adult, have there been types of exercises or puzzles that you think have improved your mathematical intelligence and in particular the spatial thinking?...
1
vote
0answers
30 views
How do we prove that $\sum_{i=1}^{n} X_{i}$ is a sufficient statistic in the present context?
Let $n$ items be drawn in order without replacement from a shipment of $N$ items of which $N\theta$ are bad. Let $X_{i} = 1$ if the $i$-th drawn is bad and $X_{i} = 0$ otherwise. Show that $\sum X_{i}$...
2
votes
1answer
36 views
Show that $\min\{X_{1},X_{2},\ldots,X_{n}\}$ is sufficient for $\mu$ when $\sigma$ is fixed
Let $X_{1},X_{2},\ldots,X_{n}$ be a sample from a population with density $p(x,\theta)$ given by
\begin{align*}
p(x,\theta) = \frac{1}{\sigma}\exp\left\{-\left(\frac{x-\mu}{\sigma}\right)\right\}
\...
0
votes
1answer
19 views
Projection of point to plane using normal equation
Say the place is parametrized as $x_1 = y_1 + y_2$ , $x_2=y_1-y_2 , x_3 = y_1+y_2$ and a point in $\mathbb{R}^3$ $x_1 = 2 , x_2 = 1 , x_3 = 3$
What would be the most efficient way to find the ...
0
votes
0answers
36 views
A coin is tossed until a head turns up and $\$f(n)$ is paid out, find the expected value of the payment
A fair coin is tossed until heads turns up for the first time. If heads turns up for the first time on the nth toss you receive $2^n$ dollars.
(a) Show that the expected value of your winnings ...
0
votes
0answers
14 views
The relation between the F1 score and the combined work formula?
I am wondering why we should use the harmonic mean(as stated in Wikipedia: "the traditional F-measure or balanced F-score (F1 score) is the harmonic mean of precision and recall") as the F1 measure, ...
0
votes
2answers
28 views
Probability question : Normal distribution
An auto parts company, produces cylinder liners for engines of 1.2 inches in average diameter with a standard deviation of 0.1 inches. Every piece has a diameter less than an inch or more than 1.4 ...
7
votes
1answer
136 views
What is the best way to study graduate level mathematics?
I am studying a 400/500 level measure theory math book on my own.
Right now, when I read it I try to read the proposition then the following proof. And then try to do the exercises on my own.
I ...
0
votes
2answers
26 views
Need help in finding bounds on inequality
Determine the condition on $|x-2|$ that will assure that
$|x^2-4|<1/2$ will work
So $|x^2-4|=|x+2||x-2$|
Do I assume that $|x-2|<1$ or $|x-2|<0.5$
I would like to see full details. Can ...
0
votes
1answer
43 views
Why $C(u_1,v_1) + C(u_2,v_2) - C(u_1,v_2) - C(u_2,v_1) \geq 0$ is equivalent to $\frac{\partial^{2}C(u,v)}{\partial u\partial v} \geq 0$?
A bivariate function $C(u,v)$ that maps $[0,1]^{2}$ to $[0,1]$ is a copula if it satisfies the following two conditions:
(i) Boundary conditions:
\begin{align*}
C(u,0) = 0\\
C(0,v) = 0\\
C(u,1) = u\\
...
1
vote
1answer
51 views
Average approximates definite integral.
In my studies, I read a result similar to the following.
Let $g:[0,1]\rightarrow \mathbb{R}$ be twice continously
differentiable on $[0,1]$ (this is probably stronger than necessary).
Then \...
0
votes
1answer
47 views
Moment Generating Function of beta ( Hard )
Given $X$ is a random variable ~ $Beta ( a , b)$ distribution and
$X$ belongs in (0,1)
Does the (MGF ) $E[e^{tx}]$ exist for every value of $a , b$ ?
(Mgf must not be equal to infinity in order ...
0
votes
2answers
44 views
How do i solve P(A\B∪C)
I just started set theory is math and with lack of understanding I've come here for some much-needed assistance. How would I go about determining what the probability is if its A given B or C?
I've ...
-1
votes
0answers
23 views
Integral Is not finite?
How can we prove with calculations that the below integral is not finite `
for $t\ge$ $\dfrac{1}{2}$
$\int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}}e^{-x^2(1/2-t)}\, dx.$
I would be pleased if you ...
0
votes
0answers
38 views
Where do I find exercises very similar to this one?
In my book Operational Research by author Hamdy A. Taha is this following exercise
A home supply store can place orders for fridges at the start of each month for immediate delivery. A cost of $\$ ...
0
votes
0answers
28 views
Rotations of Lebesgue measurable sets are Lebesgue measurable
If $E \subseteq \mathbb{R}^n$ and $Q:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is an orthogonal map, then I know that $\mu^*(QE)=\mu^*(E)$, i.e., the Lebesgue outer measure is invariant under rotations.
...
0
votes
0answers
17 views
Deriving the cumulative distribution and probability density for the following random variables
Given U is randomly chosen from [0,1] with uniform distribution, find the CDF and PDF of Y given:
a) $Y = \frac{1}{2+U}$
b) $Y = ln(U+1)$
a) $P(\frac{1}{2+U} \leq y) = P(U \geq \frac{1}{y}-...
1
vote
0answers
26 views
Algorithm to optimize redistribution of balls amongst urns
Here is the question:
Say we have k urns with 1 ball in each urn. At each iteration of the game, I pick one urn and redistribute its contents amongst other urns and each urn can receive at most one ...
0
votes
2answers
28 views
When do exponential functions form a basis?
I understand that exponential functions of the form $e^{a_{i}x}; i=1,2,..,n$ and $a_{i}s$ are real and distinct are linearly independent. Do these form the basis of vector space of continuous ...
-1
votes
1answer
25 views
Need clarification on question in Bartle’s Intro to real analysis 3rd ed
On page 85 he estimates the following for the cubic equation
$x^3-7x+2=0$
His estimate:$|x*-x_0|<\frac{3^{n-1}}{7^{n-2}}*20$
I don’t understand how the estimate for the denominator is achieved
...
0
votes
0answers
16 views
Find the profit of a store from given instructions using entropy and probability
A restaurant sells N number of different dishes (Bi where i= 1,2,3,...N). Now we assume that ingredients of each dish to be sold is independent and Assuming the estimated amount of each dish or ...
1
vote
0answers
53 views
Karlin Rubin Theorem UMP (Uniformly most powerful test ) Is it wrong?
Suppose $X_1, X_2, X_3,\ldots, X_n$ are i.i.d. random variables with a common Poisson$(\lambda)$ distribution.
$$X=(X_1, X_2, X_3,\ldots, X_n)$$
and $g(λ)=\lambda(1 - e^{-λ})$
, with $(λ>0)$
...
0
votes
1answer
18 views
Epsilon transition in NFA to DFA conversion
I worked through this conversion and it all makes sense except for one small part. Shouldn't $(q_1q_2)$ go to $q_1$ in the DFA on input $0$, not a self loop?
We have state $q_iq_2$ to begin with ...
4
votes
6answers
366 views
Solving $3x^2 - 4x -2 = 0$ by completing the square
I can't understand the solution from the textbook (Stroud & Booth's "Engineering Mathematics" on a problem that involves solving a quadratic equation by completing the square.
The equation is ...
-1
votes
0answers
27 views
How to show that $E(\hat{Y}| Y = y) = E(\theta |Y=y )$?
Show that $$E(\hat{Y}| Y = y) = E(\theta |Y=y ),$$
where $\hat{Y}$ is conditionally independent to Y given $\theta.$
Note that $E(\hat{Y}| Y = y) $ is equal to the mean of the predictive ...
2
votes
1answer
34 views
Solving simultaneous equations via $u$ and $v$
I'm self-studying from Stroud's amazing "Engineering Mathematics" textbook, and have run into an issue in one of the final end-of-chapter exercises for in the "Linear Equations" chapter.
Here's the ...
0
votes
3answers
25 views
Simplyfying each side separately and solving linear equation
I'm given an equation that first needs simplifying:
$$\frac{x-2}{x-4} - \frac{x-4}{x-6} = \frac{x-1}{x-3} - \frac{x-3}{x-5}$$
My next step is:
$$\frac{(x-2)(x-4)(x-6)}{x-4} - \frac{(x-4)(x-4)(x-6)}{...
0
votes
1answer
23 views
Why $f(x)=x^2$ is local Lipschitz, general question about local/global Lipschitz.
I'm trying to understand the difference between global lipschitz and local lipschitz.
Let $f(x)=x^2$ while $x \in \mathbb{R}$ if we look at global lipschitz, for all $M \subset \mathbb{R} \times \...
0
votes
2answers
35 views
When can we Central Limit Theorem approximation with good approximation?
I think we an use it when n(no. of trials) is large.
But my textbook used this approx. by stating that since the expectation is large, we use the approx. I'm unable to understand this, would ...
5
votes
3answers
493 views
Can't solve system of linear equations (that need simplification first)
I'm self-studying from Stroud & Booth's amazing "Engineering Mathematics", and am stuck on a problem at the end of the "Linear Equations" chapter.
I've a system of two linear equations:
$$\frac{...
0
votes
1answer
15 views
Dart board probability using line method with Poisson application
You randomly throw darts at a dartboard, one dart every second. Suppose that every dart independently hits the dartboard at distance X from the center, where X is a Unif[0,30] random variable. Your ...
0
votes
1answer
38 views
find P( X$\lt$ Y) given X and Y are two independent exponential RV
The question is as follows:
You arrive at the post office and as you enter, each of the two clerks, Jim and Jack, starts serving a client. The amount of time needed by Jim to serve his client is X ...
0
votes
1answer
21 views
Let $(X,Y)$ be a uniformly chosen point of a region $A$, given $A$ compute $EX$
Let $(X,Y)$ be a uniformly chosen point of a region $A \subset \mathbb{R}^2$
Given we have the following joint pdf:
$$ f(x,y) = \begin{cases} \dfrac{a}{ \text{area of}~ A} & (x,y)\in A \\ 0 &...
0
votes
1answer
35 views
let z1 and z2 be independent standard normal RV, find the pdf of $e^{3Z_1+2Z_2}$
Let Z1 and Z2 be independent standard normal random variables. Find the following
The probability density function of $e^{3Z_1+2Z_2}$
The given solutions is as follows:
But what doesn't make sense ...
0
votes
1answer
31 views
How to understand this probability equation?
$\mathbf { x } ( t ) = g ( \mathbf { s } ( t ) ; \xi ) + \mathbf { n } ( t )$, where n(t) denotes the noise or modeling error and ξ the parameters of mapping $g$
How to understand the following ...
1
vote
1answer
28 views
Boundary of the sum of an Infinite horizon discounted model
The sum of an infinite horizon discounted model is given as follows:
$$R_t = r_{t+1} + \gamma r_{t+2} + \gamma^2r_{t+3} + ... = \sum_{k=0}^\infty \gamma^kr_{t+k+1}.$$
As can be seen, the sum is ...
1
vote
0answers
48 views
probability that sum of geometric random variables is odd
Let $X$ and $Y$ be independent geometric random variables with the same parameter $p$. What is the probability that $X+Y$ is odd?
Based on the provided solutions, I can follow through until this part ...
