All Questions

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Random Process derived from Markov process

I have a query on a Random process derived from Markov process. I have stuck in this problem for more than 2 weeks. Let $r(t)$ be a finite-state Markov jump process described by ...
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Expected Value Problem (Q-function…inside a function)

I'm working through my textbook for a communications course I'm taking, and this problem is confusing me big time. Like always, the math questions give me the most problems. Maybe I should take the ...
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Solving an inequality of the form $x^TAx\geq0$ or $x^TAx\leq0$ is straightforward. I mean we have to check if A is positive semidefinite or negative semidefinite. But what would be the solution to the ...
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Bias of Estimator with square root of a sum of squared random variables

Got a distribution of $f_X(x;\theta) = (x/\theta^2) \exp(-x^2/2\theta^2)$ for $x \ge 0$ where the MLE is calculated as $\theta_{MLE} = \sqrt{(\sum_{i=1}^{n}x^2_i)/2n}$ So now need to find if it's ...
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How many arithmetic formula can we construct?

Given the four arithmetic operators {+, -, *, /} and four times the number 4, that is {4,4,4,4}, I would like to find a way to count how many arithmetic formula I am able to construct. I find ...
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Probability Homework Technician Question

A technician working at quality control department of a production plant is inspecting a lot of $1000$ light bulbs, among which are $20$ defective. He chooses two light bulbs randomly from the lot ...
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Poisson distribution normal approximation

6.4.18. An experimenter takes a sample of size 1 from the Poisson probability model, pX (k) = e−λλk/k!, k = 0, 1, 2, . . . , and wishes to test H0: λ=6 versus H1:λ<6 by rejecting H0 if k ≤2. (a) ...
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Polynomial interpolation when the coefficient vector has bounded 1-norm

Given $1<\alpha_{1}<\alpha_{2}<\alpha_{3}\cdots<\alpha_{N}<2$ I need to construct a degree-$L$ (with $L>N$) real polynomial $f(x)=x^{L}+\sum\limits_{i=1}^{L}b_ix^{L-i}$ which ...
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Checking Sudoku - sufficient sums

Are the following condition sufficient for checking if solution of Sudoku with (extended output) is valide : sum of values in each row, column and subsquare is equal to 45 and sum of squares of ...
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A combinatorial Identity considering Arithmetic Geometric Mean

I met the following combinatorial identities following the footsteps of Gauss in Borwein and Borwein's Pi and AGM (p.6); i.e. trying to prove the eq. (1.2.5) on this page. Prove that ...
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selecting marbles

An urn contains $r$ Red and $b$ Blue marbles. A fair coin is flipped. If the flip is Heads then $h$ Red marbles are added to the urn. If the flip is Tails then $t$ Blue marbles are add to the urn. Now ...
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An irregular 6 faced dice

An irregular 6 faced dice is such that the probability that it gives 3 even numbers in 5 throws is twice the probability that it gives 2 even numbers in 5 throws .How many sets of exactly 5 trails can ...
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Is reflection or rotation in a $2$-dimensional normed space isometric?

Is reflection in the $x$-axis or in the line $y=x$ in a $2$-dimensional normed space isometric? How about rotation through a right angle? If so, what is the proof?
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Row space and kernel in linear transformations

I am preparing a "dictionary" that translates between the "language of matrices" and the "language of linear transformations" in linear algebra. The dictionary looks more or less like this: Language ...
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Measurable Maps

Let $X$, $M$ be two metric spaces and $\nu:X\rightarrow \mathcal{M}_1(M)$, $x\mapsto \nu_x$ a map, where $\mathcal{M}_1(M)$ is the space of all probabilities over $M$ with the Boral $\sigma$-algebra, ...
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How to weight Jaccard Similarity

I'd like to calculate the similarity between two sets using Jaccard but temper the results using the relative frequency of each item within a corpus. Jaccard is defined as the magnitude of the ...
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Shortest path that has crossings?

Consider a finite set of points in a surface of dimension 2. Is there any simple example, where the shortest closed path linking them all (like a possibly non-simple polygon, but with geodesics ...
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What is the general formula for electoral districts tying.

I apologize if this question is a bit of a read. (You might want to get a frosty beverage.) Professor Alan Natapoff of MIT demonstrated, if 9 Voters are districted into 3 electoral districts of 3 ...
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Probability question involving sets of cards

I have an infinite deck built out of sets of 10 cards (in other words 10*n cards). The sets are identical so one '2' is identical to another '2'. A player draws 6 cards. If he draws: any '1' AND a ...
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Confusion related to dynamic programming

I was going through this dynamic programming problem. However, I have a confusion In the third picture, having the black border, I didn't get how the For each, we try each string of the k ...
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Estimating the number of tickets bought in a lottery

A national lottery has the format where $7$ numbers are chosen from $45$ without replacement. The first $6$ numbers chosen constitute the "winning numbers", while the last number chosen is the ...
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Bounding the power of expected value of functions of a random variable.

I am interested in a problem and I do not know where to start looking for possible similar setting. If anyone has a direction to suggest, it would be greatly appreciated. Consider a (finite) set ...
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Taylor series representation of a function.

I'm working on expressing the function $f(x)=\frac{6}{x}$ as a taylor series about $-4$. I've got the general idea, but I'm not quite there yet. I've come up with the equation ...
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Simple function converging to a smooth function almost everywhere

I'm having troubles proving following statement $$\lim_{\varepsilon \to 0^+} \sum_{i=1}^{N(\varepsilon)} f(\xi_i) \chi_{(x_{i-1},x_i)}(x) = f(x) \qquad \mathrm{a.e. on}\ [0,1].$$ $f(x)$ is a ...
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First Order PDE (Substitution)

I would like to find out what substitutions I should use to solve the following PDE: $${u_t+(x+t)u_x+u=x}$$ My professor advised that I try the substitution ${v=\ln(u)}$. However, I think that the ...
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Area of a parallelogram in R3

Let A = (2, -3, 1), B = (5, -3, -1), C = (-2, -3, 5), and D = (1, -3, 3) I have the above situation. I know this: A = || AB X AD || gives the area of the parallelogram. My end components are: 0i - ...
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Isomorphisms of CW Coverings

This is an exercise I've been working on from Hatcher (1.3.32): Consider covering spaces $p: \tilde{X} \to X$ with $\tilde{X}$ and $X$ connected CW complexes, the cells of $\tilde{X}$ projecting ...
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convergence of functions on probability measure

I am studying a problem in game theory, but I am lacking on knowledge to deal with a continuum of distribution functions convergence. $\mathfrak{F}([0,1])$ is the set of distribution functions over ...
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Euclids Algorithm Proof

Let $a,b$ be an element in $\mathbb{Z}$ with $a \ge b \ge 0$, let $d := \gcd(a,b)$ and assume $d \gt 0$. Suppose that on input $a,b$, Eculid's algorithm performs lambda division steps, and computes ...
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skinned surface

In the representation of a skinned surface using $B$-Spline, I have $K+1$ given curves of degree $p$ on a common partition $U$ and I want to construct the surface $S(u,v)$ with these curves as ...
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Uniform integrability, book

I search about this theme, in the books is as exercise. But I want some more theory. What book recommend?
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Tensor product of module homomorphisms

This is an exercise problem from Hungerford's Algebra but first I'll state a result that forms the background to the problem. Let $R$ be a ring. Let $A,A'$ be right $R$-modules. Let $B,B'$ be left ...
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Morphology of binary images

During the lecture we talked about analysis of pictures and got some exrecises. Other students say that this is very easy but I don't get a good answer. Here the facts: Suppose $A$ is a bounded ...
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Lines in $\mathbb{A}^3$

This seems intuitive, but I'm having trouble coming up with an exact matrix for the problem. Let $\{L_1, \ldots, L_N\}$ be a set of lines through the origin $(0,0,0)$ in the affine space ...
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Show that for $k \gt 0$ and $m \ge 1$, $x \equiv 1 \pmod {m^k}$ implies $x^m \equiv 1 \pmod {m^{k+1}}$.

Here's what I've done. I don't think I going the right way. If $x \equiv 1 \pmod {m^k}$, then $x^m \equiv 1^m \equiv 1 \pmod{m^k}$. $\Rightarrow x^m = 1^m + m^kn$, for some $n \in \Bbb Z$. ...
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Derivation of Grassmann valued functional

I'm trying to evaluate $$\frac{\delta}{\delta \eta(x)}e^{-\int dz \theta^*(z)\eta(z)}$$ Where $\theta^*(x)$ and $\eta(x)$ are Grassmann valued functions. The context of the functional is in term of ...
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