# All Questions

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### Can a discrete random variable be both larger and smaller than another?

Suppose we have two discrete random variables $A$ and $B$ with a finite number of possible outcomes. Then the expression '$A < B$' can be interpreted as a random variable itself, taking the values ...
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### Optimal control for hitting a random point with gaussian distribution

A particle $X$ starts from the origin $X_0=0$ of the real line and can move to the right or the left with speed $\pm 1$ and should hit a point, $\xi$, normally distributed (mean zero, and variance 1) ...
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### How to Represent a 3D Line under Polar Coordinates

In one of my applications, I need to represent a line under 3D polar coordinates system. In 2D, we can define a line by a distance to the origin and then a angle indicating the direction of the line ...
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### Explanation for the orders of subgroups and number of groups with these orders.

This was a question on my exam that was just given back and I need help understanding why both (a) and (b) and (c) are the answers they are. In each group listed below, give the orders of the ...
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### Show an iterative fixed point method does not converge

I was asked the following question: let $g(x)=\frac{30}{1+x}$. Notice that $g(5)=5$. Is there an $\epsilon >0$ such that the series $\{x_k\}_{k=0}^{\infty}$ defined by $x_{k+1}=g(x_k)$ and ...
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### Markov Chains: Limiting probabilities of positive recurrent states sum to one?

I have a question about Markov chains. I am trying to understand the proof of Proposition 2.6 of http://www.columbia.edu/~ks20/stochastic-I/stochastic-I-MCII.pdf. The setting is: we have a positive ...
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### Integral with absolut-value function

How do I seperate the following integral? The integral of $|x^2-y|$ with $|y| \leq 1$ and $|x| \leq 1$. I know that the absolute value is positive for $x^2 \geq 1$ and negative for $x^2$ but I am ...
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### Isomorphism,on ${R}^4$

I dont understand what the function is for part (a) such that a mapping from $X\in T_p{R}^4$ to $w(X,-)\in T^{\star}_p{R}^4$ be an isomorfism!. So Consider on ${R}^4=(x_1,y_1,x_2,y_2)$ the ...
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### Submultiplicativity stronger than triangle inequality?

I would like to ask a question about matrix norm. Is the submiltiplicativity property always stronger than the triangle inequality? So, if i prove for a matrix norm that it's submultiplicative, i ...
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### Concrete representation of the annihilating algebra

Suppose $\mathfrak{M} = A^{\prime\prime}$, where $A$ is a concretely described subalgebra of $\mathcal{B}(\ell^{2}(\mathbb{N}))$. In some instances, it is possible to provide a concrete description of ...
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### Extending a polynomial function on an interval to be infinitely differentiable on all of R

If $f:(a,b) \to \mathbb{R}$ is a polynomial function, can it be extended to $g:\mathbb{R} \to \mathbb{R}$ such that g is infinitely many times differentiable and it is NOT the same polynomial? What ...
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I need help getting a proof, I don't want solution to the problem just help guiding me to complete the proof. Suppose $m,n$ are coprime, Prove that $a \equiv b \mod{mn}$ if and only if: $a ... 2answers 103 views ### Finding the second derivative; What am I doing wrong? Original Question:$xy+y-x=1$Find the second derivative;$d^2y\over{dx^2}$$(xy+y-x=1) We are allowed to use either notation as far as I know: {dy\over{dx}} or {y'}. Because ... 3answers 34 views ### Coin and steps -Probabilty and Statistics Turn a coin and if it falls heads move three places to the right otherwise move 2 places left. After 20 times you turn the coin, in what positions might you be and what is the probability to be in ... 2answers 268 views ### Closedness and convexity of half spaces \mathbb{R}^n determined by hyperplanes Every hyperplane divides \mathbb{R}^n into two "half space": the set of points "on and above" the hyperplace, H^+ = \{ \mathbf{x} \mid \mathbf{a} \cdot \mathbf{x} \geq \alpha \}, and the set of ... 3answers 72 views ### Why does aH = Ha \neq ah = ha For normal subgroups...How come aH = Ha does not imply that ah = ha for all h \in H? I'm showing that every subgroup of a commutative group is normal and I thought I had it on that one but ... 1answer 58 views ### Nonconvex set converging to a convex set despite holes I'm looking at the example in Figure 4-7 of "Variational Analysis" (Rockafellar and Wets). Basically, there's a sequence of sets C_{\nu} riddled with holes, and it states that the sequence ... 4answers 120 views ### How to show that x_n=-\sqrt{n} + n\ln\Big(1+\frac{1}{\sqrt{n}}\Big) is decreasing? I am a non-mathematician who knows some elemententary calculus ans I want to prove that the sequence (x_n) given by$$ x_n=-\sqrt{n} + n\ln\Big(1+\frac{1}{\sqrt{n}}\Big)  is decreasing. Is there ...
I have this line integral: $\oint 3ydx+x^2dy$ and the path is a line from $(0, 0)$ to $(1, 0)$ (so this is $y=0$), another line from $(1, 0)$ to $(1, 1)$ (so this is $x=1$) and a curve $y=x^2$ from ...