Recent Questions - Mathematics Stack Exchange |
- How to interpret the periodic summation formula?
- $\mathcal{H}$ is Hilbert space. A finite subtset of vectors $S = \{x_1, x_2, ..., x_n\}$ must have at least two identical vectors if:
- Divergence Question Verification
- Find two Matrices that are 'almost' similar.
- Is my derivation right?
- Strong separation of two disjoint nonempty closed convex sets, one of which is compact
- Optimization with a control comes from Borel regular measure
- Move four wheels on a curved path
- Efficient way to test 0 probability event from a degenerated multivariate Bernoulli distribution
- Understanding Euler's Identity (complex)
- Maximizing and minimizing a function with x and y
- Determine the image of the unit circle $S^1$ by the action of the matrix $e^A$.
- How exactly do I 'show' that an equation represents something?
- Evaluating a line integral clockwise around the perimeter of a triangle of area 3
- How to solve $Cx^2 y'' + xy'- y = 0$?
- Evaluate the line integral along a parabola
- How to converts total days passed since epoch from real Gregorian calendar to teorical Gregorian that all years are leap years?
- Determine the value of k so that the following system is consistent.find solution for these values of k.
- Formula for function
- Sum of two measurable functions is measurable
- Bounding a convergent sum
- How to prove the following inequality between $u$ and integral of its gradient
- Find solution of a Polynomial Equation [closed]
- Relation between inverse of truncated and original matrix
- How to calculate the probability of the difference between two negative binomial distribution?
- Statistical analysis to define homogeneity and heterogeneity of a population
- Cross Products and charged particles in magnetic fields.
- Selecting the Real Analysis Textbooks
- How to use Hensel Lifting to solve $x^2\equiv 2\pmod {17^3}$?
- Definition of a countable set
| How to interpret the periodic summation formula? Posted: 07 Jun 2022 05:56 PM PDT The formula stated here: https://en.m.wikipedia.org/wiki/Periodic_summation is not clear for me what the point of such a formula is and if i interpret it correctly. For me, this infinite sum, for a given $t$ will mostly sum zeroes and just add one $s(t)$ of the aperiodic function $s(t)$. so why not write $s_P(t) = s(t)$ and that's it...? Since i guess that $s$ being aperiodic, then $s(t+nP) = 0$ for $n>0$? Because $s$ without the subscript P is aperiodic i.e. non periodic! Or am I interpreting it wrong (i guess i do but why the hell don't they explain it clearly?!) I would rather imagine that it would be something like this infinite sum "placing" the value of $s(t)$ at integer multiples of the period $P$ (a bit like the dirac that fires when its arg is ==0)? But if it is so please can someone indicate how this can ne inferred from the formula... since $s$ is aperiodic (and only $s_P$ is periodic)? |
| Posted: 07 Jun 2022 05:49 PM PDT for any vector $\xi \in \mathcal{H}$, there exist two vectors $x_i, x_k$ in the set $S$ such that $\langle x_i, \xi \rangle = \langle x_k, \xi \rangle$. (two such vectors may depend on $\xi$) If the set S satisfies the above property, are there two identical vectors in it? I have been trying to construct a counterexample, but I didn't make it. Moreover, is this question easy to solve when the Hilbert space is separable? |
| Divergence Question Verification Posted: 07 Jun 2022 05:45 PM PDT can someone please do this and just provide the final answer, I want to see if I am doing this correctly:) Evaluate the surface integral (doubleintegral F dot dS) where the vector field is F(x,y,z)=<xz,-2y,3x> and S is the surface of the region bounded above by the paraboloid z = 4 - x^2 -y^2 and below by the plane z = 0. |
| Find two Matrices that are 'almost' similar. Posted: 07 Jun 2022 05:44 PM PDT Let, $J_{k}(\lambda)$ be the Jordan block of size $k$ and eigen value $\lambda$. Then we define, $$M = J_{5}(0) \oplus J_{2}(0) \oplus J_{2}(0),$$ and $$M' = J_{5}(0) \oplus J_{1}(0) \oplus J_{3}(0).$$ Observe that, $M$ and $M'$ have the same characteristic polynomial, same eigen values with the same multiplicities, and the same minimal polynomial, but they are not similar because they do not have the same Jordan Canonical Form. Is my reasoning correct? Thank you! |
| Posted: 07 Jun 2022 05:42 PM PDT $\theta$ is a random variable whose prior distribution is a normal distribution with mean $m_0$ and variance $\sigma_0^2$. Let $e \in \mathbb{R}^k$, $e=\left(1,..,1\right)^T$, $x \in \mathbb{R}^k$, and $x=\theta \cdot e+u$, where $u$ follows normal distribution with mean $0$ and variance matrix $\Sigma$. This can be interpreted as receiving $k$ unbiased signals. Given $x$, the posterior distribution over $\theta$ follows a normal distribution with variance $\sigma_*^2=\left(\sigma_0^2+e^T \Sigma^{-1} e\right)^{-1}$ and mean $m_*=\left(\sigma_0^{-2}m_0+e^T\Sigma^{-1}x\right) \sigma_*^2$. Now I would like to derive the distribution over the posterior distribution of $\theta$. Note $\sigma_*^2$ does not depend on $x$. So I only need to derive the distribution of $m_*$. Let $J$ be a $k\times k$ matrix whose all elements are $1$. It follows a normal distribution with mean $m_0$ and variance $\sigma_*^4\left(\sigma_0^2e^T \Sigma^{-1}J\Sigma^{-1}e+e^T\Sigma^{-1}e\right)$. Note $e^T\Sigma^{-1}e$ indicates the sum of all elements in $\Sigma^{-1}$ while $e^T \Sigma^{-1}J\Sigma^{-1}e$ indicates the square of the sum of all elements in $\Sigma^{-1}$. As a result, no matter what value $k$ is, as long as the sum of all elements of $\Sigma^{-1}$ is fixed, the distribution over the posterior distribution of $\theta$ is fixed. In other words, the informativeness of signals is fixed. I like this neat result. I was wondering whether my derivation is correct and whether it is a well known result in probability theory. Thank you very much! |
| Strong separation of two disjoint nonempty closed convex sets, one of which is compact Posted: 07 Jun 2022 05:39 PM PDT I'm trying to prove below theorem from this lecture note. Could you have a check if my proof is fine? Thank you so much for your help!
My attempt: We need the following result. Lemma 1: Let $A$ and $B$ be two nonempty disjoint closed subsets of a t.v.s. $X$ such that $A$ is compact. Then there exists a neighborhood $V$ of $0$ such that $(A+V) \cap B=\emptyset$. Lemma 2: Let $A$ and $B$ be two nonempty convex subsets of a vector space $X$. If $\operatorname{aint} (A) \neq \emptyset$ and $B \cap \operatorname{aint} (A)=\emptyset$, then $A, B$ can be separated by a hyperplane. Here $\operatorname{aint} (A)$ is the algebraic interior of $A$. If, moreover, $X$ is a t.v.s., $\operatorname{int}(A) \neq \emptyset$ and $B \cap \operatorname{int}(A)=\emptyset$, then $A$ and $B$ can be separated by a closed hyperplane. Lemma 3: Let $A$ be a convex subset of a vector space $X$ such that $\operatorname{aint} (A) \neq \emptyset$. Here $\operatorname{aint} (A)$ is the algebraic interior of $A$. Let $\alpha \in \mathbb R$ and $\ell: X \to \mathbb R$ be a nontrivial linear functional. Then the following statements are equivalent.
By Lemma 1, there is a neighborhood $V$ of $0$ such that $(A+V) \cap B=\emptyset$. Because $X$ is locally convex, we can assume $V$ is convex. Let $A' := A+V$. Then $A'$ is convex and open. WLOG, we can assume $0 \in A$. Then $V \subset A'$ and thus $\operatorname{int}(A') \neq \emptyset$. By Lemma 2, there is $\ell \in X^*$ such that $\sup \ell(A') \le \inf \ell (B)$, i.e., $$ \ell(a) \le \inf \ell (B) \quad \forall a \in A'. $$ Notice that $A'=\operatorname{int}(A') \subset \operatorname{aint}(A') \subset A'$, so by Lemma 3, $$ \ell(a) < \inf \ell (B) \quad \forall a \in A'. $$ Because $0 \in V$, we get $A \subset A'$. So $$ \ell(a) < \inf \ell (B) \quad \forall a \in A. $$ Notice that $A$ is compact and $\ell$ continuous, so $$ \sup \ell(A) = \max \ell(A) < \inf \ell(B). $$ This completes the proof. |
| Optimization with a control comes from Borel regular measure Posted: 07 Jun 2022 05:35 PM PDT I have a task in optimization to do. That is \begin{align} \min_{z} \hat{J}(z):=\int_Q |u(z)-u_d|(z) \end{align} such that \begin{align} &u_t-\Delta u=f\ \ \ \ on \ \ Q \\ &\partial u_n=0 \ \ \ \ on \ \Sigma \\ & u=z \ \ \ \ in \ \ \Omega \end{align} where $$z \in \mathcal{Z}=\{z \in \mathcal{M(\bar{\Omega}): \|u\| \leq \alpha,\ \alpha>0}\}$$ and $Q=\Omega \times (0,T),\ \ \Sigma= \partial \Omega \times (0,T)$, $\partial \Omega$ refers to the boundary of $\Omega$.\ My question is how is that $u=z$ in $\Omega$? By definition of $\mathcal{M}(\bar{\Omega})$ that I google, $z \in \mathcal{M}(\bar{\Omega})$ means it is a measure that is defined on $B(\bar{\Omega})$ where the latter notation means the Borel field of $\Omega$ which consists of subsets of $\bar{\Omega}$, so logically $z$ is not defined on $x$ it is defined on $\{x\}$ which contradicts what I have in the above pde. So could anyone explain how the elements of $\mathcal{M}(\bar{\Omega})$, and how they are defined on $\Omega$? |
| Move four wheels on a curved path Posted: 07 Jun 2022 05:33 PM PDT I'm researching the movement of four wheels on a curved track. The image below is of a stairlift.
As it is known, the four moving wheels are in a fixed position relative to each other. This structure can follow a direct and curved path. but how? As shown below, this should not be possible on a curved path. When two wheels are tangential to the direction of track, the other two wheels are in the inappropriate position and it is not possible to move.
But according to the first picture, in practice this is possible. But I do not know how |
| Efficient way to test 0 probability event from a degenerated multivariate Bernoulli distribution Posted: 07 Jun 2022 05:30 PM PDT Given a multivariate normal distribution $X \sim N(\vec{0}, I), 0 \in \mathbb{R}^d, I \in \mathbb{R}^{d \times d}$, and some low rank matrix $A \in \mathbb{R}^{d\times d}$. I can easily generate a new random variable $Y := AX$ where $Y \sim N(\vec{0}, AA^T)$. Let $Z = sign(Y)$, $sign$ is the sign operation. It would follow a degenerated multivariate Bernoulli distribution. My question is given $A$ and some random binary vector $z \in \{-,+\}^d$, how can I test whether it has 0 probability in distribution $Z$? |
| Understanding Euler's Identity (complex) Posted: 07 Jun 2022 05:44 PM PDT For a complex number $z=a+bi$ and a positive real value $R$, we have $e^{Rbi}=\cos(Rb)+i\sin(Rb)$. I am struggling to understand this since no matter how large $b$ or $R$ is, we have $|e^{Rbi}| \in [-1, 1]$. What is the best way to understand this intuitively? For instance, in an applied sense, is it true that $$\bigg|\sum_{z: \ a, b \geq 0}e^{Rbi}f(z)\bigg|\leq \bigg|\sum_{z: \ a, b \geq 0}f(z)\bigg|,$$ where $f$ is some generic function and $\sum_{z: \ a, b \geq 0}f(z)>0$? This makes sense to me because each $|e^{Rbi}|$ is no larger than $1$. |
| Maximizing and minimizing a function with x and y Posted: 07 Jun 2022 05:22 PM PDT It is clear that 253x + 256y = 253(x+y) + 3y. For a pair of integers x and y satisfying: $$253x + 256y = 1$$ The absolute value of x is minimum. Then, x = ? and y = ? I tried squaring the equation to make it a quadratic, and seeing if I can make a graph of it from there, but that didn't work as my variables always cancelled out. I thought about trying to do a calculus maximization/minimization but I have no idea on how to even start when all I'm given is two of the exact same equation. All help is appreciated! The answers are x = 85 and y = -84 |
| Determine the image of the unit circle $S^1$ by the action of the matrix $e^A$. Posted: 07 Jun 2022 05:35 PM PDT We have: $e^{ \begin{pmatrix} -5 & 9\\ -4 & 7 \end{pmatrix} }$ I need to determine the image of the unit circle $S^1$ by the action of the matrix $e^A$. I think that I know how to calculate $e^A$: I get the Jordan decomposition: $$A = \begin{pmatrix} -5 & 9\\ -4 & 7 \end{pmatrix} = \begin{pmatrix} -6 & 1\\ -4 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix} \cdot \frac{1}{4} \begin{pmatrix} 0 & -1\\ 1 & -6 \end{pmatrix} $$ With eigenvalues: $\lambda$ = 1, algebraic multiplicity = 2, eigenvecotrs: $\left\{ \begin{pmatrix} 1\\ 0 \end{pmatrix}, \begin{pmatrix} 0\\ 1 \end{pmatrix} \right\}$ $$ \displaystyle e^A = \sum^{\infty}_{i = 0} \frac{A^i}{i!}$$ $$e^A = \begin{pmatrix} -6 & 1\\ -4 & 0 \end{pmatrix} \cdot \left( \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} + \displaystyle \sum^{\infty}_{i = 1} \frac{ \begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix}}{i!} \right) \cdot \frac{1}{4} \begin{pmatrix} 0 & -1\\ 1 & -6 \end{pmatrix}$$ $$e^A = \begin{pmatrix} -5 & 9\\ -4 & 7 \end{pmatrix} = \begin{pmatrix} -6 & 1\\ -4 & 0 \end{pmatrix} \cdot \begin{pmatrix} \displaystyle \sum^{\infty}_{i = 1} \frac{1}{i!}& \displaystyle \sum^{\infty}_{i = 1} \frac{2^{i-1}}{i!}\\ 0 & \displaystyle \sum^{\infty}_{i = 1} \frac{1}{i!} \end{pmatrix} \cdot \frac{1}{4} \begin{pmatrix} 0 & -1\\ 1 & -6 \end{pmatrix}$$ Where: $$\displaystyle \sum^{\infty}_{i = 1} \frac{2^{i-1}}{i!} = \frac{1}{2} \sum^{\infty}_{i = 1} \frac{2^{i}}{i!} = \frac{1}{2}(e^2 - 1) $$ So: $$e^A = \begin{pmatrix} -6 & 1\\ -4 & 0 \end{pmatrix} \cdot \begin{pmatrix} e & \displaystyle \frac{e^2}{2} - \displaystyle \frac{1}{2}\\ 0 & e \end{pmatrix} \cdot \frac{1}{4} \begin{pmatrix} 0 & -1\\ 1 & -6 \end{pmatrix} = \begin{pmatrix} \displaystyle \frac{-3e^2 + e + 3}{4} & \displaystyle \frac{9e^2 - 9}{2}\\ \displaystyle \frac{-e^2 + 1}{2} & 3e^2 + e - 3 \end{pmatrix} $$ Now, I don't know if I did it correctly up to this point and what I should do next - to operate on my unit circle. |
| How exactly do I 'show' that an equation represents something? Posted: 07 Jun 2022 05:36 PM PDT Here is an equation and I want to know how I can show that this represents an ellipse ($a$ and $b$ are complex constants and $\alpha$ is a real variable): $$ z = ae^{i\alpha} + be^{-i\alpha} $$ I wanna learn how a mathematician would approach a problem like this, like where would you start for example. I have no idea where to even begin with problems like these. |
| Evaluating a line integral clockwise around the perimeter of a triangle of area 3 Posted: 07 Jun 2022 05:38 PM PDT I have not learned Green's Theorem so I don't think that would be applicable for me in this question. Since I'm not given actual vertices, I can't break up the triangle into line segments and parametrize separately. Is it sufficient to say that the answer is 0 because as long as you go back to where you started for a scalar function, it will be 0? Also, $f(x,y)=-y+x$ |
| How to solve $Cx^2 y'' + xy'- y = 0$? Posted: 07 Jun 2022 05:57 PM PDT How to solve the differential equation $Cx^2 y'' + xy'- y = 0$, if $C$ is positive? Attempt: I use power series and let $$y = \sum_{n=0}^{\infty} a_n x^{n+c}$$ be the solution. Getting the first and second derivatives and substituting these into the given DE, I obtained $$\sum_{n=0}^{\infty} [C(n+c)(n+c-1)] a_n x^{n+c} + \sum_{n=0}^{\infty} (n+c) a_n x^{n+c} - \sum_{n=0}^{\infty} a_n x^{n+c} =0$$ What should we continue here? If this is not the optimal solution or approach, what would you recommend? |
| Evaluate the line integral along a parabola Posted: 07 Jun 2022 05:36 PM PDT
I need help trying to find a good parameterization for this because what I've done just lands me in a mess. My work so far: Let $x=t, y=2(t+1)^2$ $$ \begin{split} r(t) &= \left<t,2(t+1)^2\right>, \quad -1\leq t \leq 0 \\ r'(t) &= \left<1,4(t+1)\right>, \quad -1\leq t \leq 0 \\ \|r'(t)\| &= \sqrt{1+16(t+1)^2} \\ -y+x &= -2(t+1)^2+t\\ &=-2t^2-3t-2\\ \end{split} $$ So we have $$ \int_0^{-1}\left(-2t^2-3t-2\right)\sqrt{1+16(t+1)^2}dt $$ This integral is really ugly. so then I tried a different method: $$ \begin{split} y &= 2(x+1)^2 \\ \frac{dy}{dx} &= 4x+4 \\ dS &= \sqrt{1+(4x+4)^2} dx\\ &=\sqrt{16x^2+32x+17} dx \end{split} $$ So we get $$\int_0^{-1} \left(-2(x+1)^2+x\right)\sqrt{16x^2+32x+17}dx$$ Again, very ugly. Can someone please help me solve this? |
| Posted: 07 Jun 2022 05:57 PM PDT Assuming:
Given a date $d$ (valid date of realistic) and assuming $d_r$ that is "days lasted from epoch date to $d$ in realistic Gregorian calendar" and $d_h$ that is "days lasted from epoch date to $d$ in hypothetical Gregorian calendar", find the formula(s) that converts $d_r$ to $d_h$. Can be like $t_1 = f_1(d_r)$ setting $f_1,f_2,...,f_n,f_{n+1}$. Don't use components of $d$ to calculate. I ask to avoid modular arithmetic and use Euclidean division notations like $\lfloor\frac{n}{d}\rfloor$ or $q(n,d)$ or $q(n/d)$ and $n-d*\lfloor\frac{n}{d}\rfloor$ or $r(n,d)$ or $r(n/d)$. If you do more:
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| Posted: 07 Jun 2022 05:14 PM PDT $[(3w+2x+4y+0z=3),(0w+x+y+z=k),(5w+4x+6y+0z=15)]$. I first used elementary row operation. Then $k$ seems to be taking any value for system to be consistent. I'm not sure of its correct and if it is how do l find for the values of $k$. |
| Posted: 07 Jun 2022 05:49 PM PDT First of all post does not represent the tag, I needed to put popular tag for submitting. I am a math self learner and I don't know the exact language of math. I try to learn calculus by using Gilbert Strang's book "Calculus". There is an interesting problem that I want to solve, but I can't find a solution. This problem is about how to represent functions using "j" letter. For example I have this f function outputs: 0 1 0 1 0 1... and I want to represent function of f by using the j, which is a number that tells which number I am looking at. So first f-0 would be j-0, second f-1 will be j-1, third f-0 will be j-2, fourth f-1 will be j-3 etc. I have found that if j is odd, then f is 1 and if j is even, then f=0. So I came up with this two functions f(j) = j^0 and f(2j) = j-j (it is really hard to represent my notebook writing in web text, those j and 2j in round brackets represent that they are written under f). So how can I represent this two functions as one function? Something like: f={0 if j=even number, 1 if j=odd number I hope I explained the problem pretty well, so you have understood it. |
| Sum of two measurable functions is measurable Posted: 07 Jun 2022 05:24 PM PDT I am trying to understand the proof of Royden when $f$ and $g$ are measurable functions, then $f+g$ is measurable. For $x \in E$ if $f(x)+g(x) < c$, then $f(x) < c - g(x)$. By the density of the rational numbers, there exists $q \in \mathbb{Q}$ such that $f(x) < q < c - g(x)$. Hence, $\{x \in E : f(x) + g(x)< c\} = \displaystyle\bigcup_{q \in \mathbb{Q}} [ \{x \in E: g(x)<c-q\}\cap\{x \in E: f(x)<q\}]$. I understand almost everything, but I don't understand why we need to use the $\bigcup$ of rational numbers. i.e., I understand that we need to use a rational number in the equation to proof each part is measurable. But why to use the union of the sets that meet the conditions for every rational number? Is not enough to meet just one national number? |
| Posted: 07 Jun 2022 05:37 PM PDT Let $\lg = \log_2$ and $n$ be a positive integer. I am trying to find an upper bound on the growth rate of the following sum: $$S_n=\sum_{i=1}^{n} \frac{1}{2^ii}e^{-n/2^i}$$ Clearly the sum converges, and furthermore it seems like $\lim_{n\to \infty}S_n = 0$. It seems like the terms achieve a maximum around $i\approx \lg(n)$ and most other terms are much smaller (i.e., don't contribute much to the sum). Given this, it seems like we have $$S_n \leq c\frac{1}{n\lg n}$$ for some $c>0$. If we were to omit the factor of $i$ in the denominator, then the sum could be easily bounded by integrating $e^{-nx}$ over a suitable interval. Similarly, if we were to drop the factor of $e^{-n/2^i}$, then we could bound the sum by the closed form for $\sum\limits_{i=1}^\infty \frac{x^i}{i} \bigg|^{x=\frac{1}{2}}$. But with both of them together, I am not sure how to proceed. |
| How to prove the following inequality between $u$ and integral of its gradient Posted: 07 Jun 2022 05:38 PM PDT I'm reading a paper and it states that the following result is well known. I can't prove it myself but would like to see a proof before I continue the reading, can anybody help?
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| Find solution of a Polynomial Equation [closed] Posted: 07 Jun 2022 05:43 PM PDT If $4b^2+1/b^2=16$ then how do I find the solution of $b^4+4/b^4-63/b^2$? From $4b^2+1/b^2=16$, I got $$(2b+1/b)^2 = 12 \tag{1}$$ and $$(2b-1/b)^2 = 20 \tag{2}.$$ By solving equation (1), $$2b+1/b = 2\sqrt{3}$$ and by solving equation (2), $$2b-1/b = 2\sqrt{5},$$ but I couldn't find a way to proceed further. 4 options are given below. From these 4, I have to choose one as answer. a) -1/4 b) -2 c) 3 d) 1/4 [Edited] Thank you all for the suggestions you made. Now I have found the answer. The following is how I came across the solution. Solution:
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| Relation between inverse of truncated and original matrix Posted: 07 Jun 2022 05:47 PM PDT I've got matrix A: \begin{bmatrix}0&0&0.9070&0.4724&0\\0.2740&0.8045&0&0&0.7579\\0.7292& 0.6490&0&0&0.5162\\0&0&0.3600&0.0949&0\\0.4633&0.3413&0&0&0.7306\end{bmatrix} with inv(A): \begin{bmatrix}0&-1.5372&1.6981&0&0.3949\\0&1.5148&0.7787&0&-2.1214\\-1.1309&0&0&5.6272&0\\4.2879&0&0&-10.8036&0\\0&0.2674&-1.4406&0&2.1091\end{bmatrix} If I remove rows(1,4) and cols(3,4) from A, it gives matrix B: \begin{bmatrix}0.2740&0.8045&0.7579\\0.7292&0.6490&0.5162\\0.4633&0.3413&0.7306\end{bmatrix} with inv(B): \begin{bmatrix}-1.5372&1.6981&0.3949\\1.5148&0.7787&-2.1214\\0.2674&-1.4406&2.1091\end{bmatrix} It is obvious that I can get inv(B) from inv(A) by: inv(B) = inv(A) (unremovedCols,unremovedRows) = inv(A)([1,2,5],[2,3,5]) What principle/theorem is behind this observation? |
| How to calculate the probability of the difference between two negative binomial distribution? Posted: 07 Jun 2022 05:16 PM PDT This was asked here: Understanding the solution of a probability question
There were various solutions that I understand. However, I wanted to know why it could not be calculated this way: Calculate the mean of each distribution. $ \mu$ for Rahul: $ \frac {.9}{.1} = 9 $ Variance for Rahul: $ \frac {.9}{.1^2} = 90 $ $ \mu$ for Toby : $ \frac {.8}{.2} = 4 $ Variance for Toby : $ \frac {.8}{.2^2} = 20 $ The joint variance is 110 -> joint standard deviation is 10.488 Expected difference of policies between Rahul and Toby is 9 - 4 = 5. Therefore, the probability that Rahul examines fewer is the probability that the difference is 0 or less. This should be the probability that the difference is less than 5/10.488 (Z = -.4767) standard deviations. With a z-score calculator this corresponds to a probability of .31679. The actual answer was .2857. Why the discrepancy in answers? Why is it inappropriate to calculate joint variance and find the probability of the difference of the expected values of each distribution? |
| Statistical analysis to define homogeneity and heterogeneity of a population Posted: 07 Jun 2022 05:21 PM PDT I am seeking a statistical test for deciding if the studied data are homogeneous or not. For instance an image region is considered to be heterogeneous if there is an abrupt change of intensity. Some techniques such as the coefficient of variation was found to be not robust. In fact, could someone please told me what are other techniques used in the same sense. EDIT : In response to this answer, as I am working with data representing intensities of pixels within an image. The heterogeneity reflects a sudden change in the intensity within a region as shown in the images below (the fully black is considered homogeneous while the other black region has a sudden change in the intensity). What I am looking for, is a metric that in somehow could make the decision (a binary decision). Does the Gini coefficient has a threshold value commonly used?
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| Cross Products and charged particles in magnetic fields. Posted: 07 Jun 2022 05:14 PM PDT Background : I am Fresher Mathematics Student in University. We are currently learning cross products. I am struggling with this question. Questions
My Attempt: I am aware that the result of the cross product of two vectors is such a vector that if I dot multiply with the either of the original vectors I get zero, Which means the cross product is perpendicular. I don't know how using that example how I can solve the problem. I would appreciate any help in the questions. Thanks! |
| Selecting the Real Analysis Textbooks Posted: 07 Jun 2022 05:14 PM PDT I am a sophomore in the US with double majors in mathematics and microbiology. I am interested in self-studying real analysis since it will help me with my current research in computational microbiology, prepare for upcoming math research (starting this Fall) on analytic number theory, and prepare for the real analysis course I will take this Fall and Putnam competition. I just finished Calculus with Analytic Geometry by G. Simmons, How to Prove It by Daniel Velleman, and How to Solve It by G. Polya. I also read some portions of Apostol's Calculus Vol. I to get a deeper view on calculus theories. (I was originally planning to read Apostol's Calculus Vol. I and Spivak's Calculus first, but I think it would be a better idea to start with real analysis since it covers all the ideas in those "advanced calculus" textbooks and much more.) My current plan is to start with one "dumbed-down" real analysis textbook and one "comprehensive, detailed, and intermediate" textbook, and advance into Rudin's Principles of Mathematical Analysis (required textbook for my real analysis course) starting this Summer, and use it in accordance with other real analysis textbooks. Could you help me on selecting one book from each category? Elementary Real Analysis textbooks:
Intermediate, detailed Real Analysis textbooks:
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| How to use Hensel Lifting to solve $x^2\equiv 2\pmod {17^3}$? Posted: 07 Jun 2022 05:34 PM PDT I got $x^2\equiv 2\pmod {17} \iff x\equiv\pm \sqrt{2} \pmod {17}$. Then, $x^2 \equiv 2\pmod {289}$. $(\sqrt 2 + 17y)^2\equiv 2\pmod {289}$ Then I got $34\sqrt 2\equiv 0\pmod {289}$ I am stuck from this part! Can someone please help me out with that. |
| Posted: 07 Jun 2022 05:21 PM PDT What is the proper definition of a Countable Set? |
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