Riccardo Sven Risuleo
  • Member for 4 years, 11 months
  • Last seen more than a month ago
  • Stockholm, Sweden
Why are $10$-sided dice not bipyramids?
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10 votes

A pentagonal bipyramid would work fine. The problem is that reading the result would be difficult. Dice roll on a surface and land on one of the faces, then you read the result (usually) on the face ...

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Geometric argument showing that the sum of the series $\sum_{n = 1}^{\infty} n^{-1}(n + 1)^{-1}$ is $1$
9 votes

Write out the partial sums of the series into a telescopic sum: $$ s_n = \sum_{i=1}^n \frac{1}{i} - \frac{1}{i+1}$$ from here we already see that $$ s_n = 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} +...

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Beaver lodges and burrows
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6 votes

I'm attempting to solve for one of the beavers, just to see if I've figured it out :) All in all, it is a graph traversal problem which you can solve with backtracking. I believe that he problem is ...

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Is there a way to preform this integral such that the answer is $e^{-|y|}$?
5 votes

Here is my attempt, it builds on Cauchy's integral theorem and requires complex numbers, but it works :) Let $$ f(y,x) = \frac{1}{\pi}\frac{\mathrm{e}^{-\frac{|y|}{2}}\mathrm e^{y \frac{i x}{2}}}{1 + ...

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How should we think of the commutator of two permutations?
4 votes

The commutator of two transformations $A$ and $B$ represents the difference between applying first $A$ then $B$ and first $B$ then $A$. In linear vector spaces it's pretty easy to follow. Imagine ...

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Intuition: Null Space being Subspace of Domain
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3 votes

The null space is the space of vectors (of the domain) that are mapped by the operator $T$ into the zero vector (of the codomain) . So, by definition, the null space has to be a subset of the domain ...

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Intuitive Interpretation of Filtration
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3 votes

The concept of filtration is required to give a formal definition of conditional expectation. In particular, conditional expectation is a random variable because of the sigma algebra of the ...

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Newton Method to find the Maximum value
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3 votes

That's because it depends a bit on which Newton method you refer to. In the one case, it's Newton's root-finding algorithm applied to the gradient of the function: this method will find a local ...

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Why do we care about the general outer product $\mathbf u\otimes\mathbf v=\mathbf u\mathbf v^{\sf T}$?
3 votes

The outer product $u v^T$ more often than not appears as an operator; for instance, as Botond pointed out, if $u$ is a unit vector, then $P_u = uu^T$ is the projector onto the $u$. There are, however, ...

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Show using the definition of the limit: $\lim_{(x,y)\to (0,0)} \frac{x^4+x^2+y^2+y^4}{x^2+y^2}=1$
3 votes

Hint: $$\frac{x^4+ y^4}{x^2+y^2} \leq \frac{x^4 +2x^2y^2+ y^4}{x^2+y^2} = x^2 + y^2.$$

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How to find MLE of this piecewise pdf?
3 votes

Your expression of the p.d.f. is not very convenient for the estimation. It's better to use $$ f_\theta(x) = \theta^{I(0 \leq x \leq 1)}(1-\theta)^{I( 1 < x \leq 2)}.$$ If you spend two minutes ...

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For a possibly negative random variable, what does $0$ expectation and an expression for the random variable that is true $\mathbb{P}$ a.s. imply?
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3 votes

Let $\Omega_1 = \{ \omega | X(\omega) = Y(\omega) - Z(\omega)\}$ and $\Omega_2$ be the complement of $\Omega_1$ in $\Omega$. Then $\mathbf P(\Omega_1) = 1$ and $\mathbf P(\Omega_2) = 0$. Then we have ...

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What are autonomous and non-autonomous systems??
3 votes

I assume you refer to dynamical systems; that is, differential equations of the form $$ \dot x = f(x,t,u).$$ These are classified according to which terms appear in $f(x,t,u)$: Time invariant if $f(...

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Are there non-standard roads to research available for gifted students?
3 votes

In the real world, saying that you have a special talent is equivalent to stating, in math, that you have a result. You need a proof. Flawed as it may be, a standardized university curriculum is proof ...

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About a spectral norm estimation
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3 votes

Yes you can. One way is to rewrite $Y$ as the product $$Y = (I_k \otimes x') \begin{bmatrix} A_1 \\ \vdots \\ A_k\end{bmatrix}$$ Then you can use norm inequalities: $$\|Y\| = \left\|(I_k \otimes x') ...

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Solving exponential integral with two variables (x and r)
3 votes

\Use the replacement $\log x = t$, then $x = \mathrm{e}^t$ and $dt = dx/x$, so your integral becomes $$ \int_{-\infty}^\infty e^{r t} \frac{e^{-t^2/2}}{\sqrt{2\pi}} dt $$ In which you should ...

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Clarification about positive semidefiniteness
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3 votes

The problem is not in $A$ and $B$, but in their product $AB$: the product is not symmetric; hence, there is no clear definition of positive definiteness. In standard parlance, a Hermitian (or ...

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Median and mode point estimator for simulation
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2 votes

The median is the value such that half of the density of above and half is below it. To find the empirical median you put the data in a long vector. Then you sort the vector. The element in the ...

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Why in SDE, we always consider the density $p(x,t|x_0)$ and never $p(x,t)$?
2 votes

The stochastic differential equation $$ \mathrm d X_t = \mu(X_t) \mathrm d t + \sigma(X_t) \mathrm d B_t,$$ is just an informal way to represent the integral equation $$ X_{t+h} - X_t = \int_t^{t+h} \...

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How do you find the distance distribution of a polygon?
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2 votes

Let $f(x,y)$ be the mass distribution of the object (the density of the object at point $x,y$). Change the coordinates to polar coordinates: $$g(\rho,\theta) = \rho f(\rho \cos\theta, \rho \sin\theta)...

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Three planes go thru a point (linear systems)
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2 votes

The equation of a plane passing through a point $(x_0,y_0,z_0)$ is $$ a(x-x_0) + b (y-y_0) + c(z-z_0) = 0.$$ Different values of the coefficients $a$, $b$, and $c$ give different planes. Pick any ...

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Does equivariance of the MLE require the function be invertible?
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2 votes

No it is not necessary that $g$ is invertible. See page 320 of Casella and Berger. The proof of the property is, however, much simpler when $g$ is invertible.

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Define the events in order to use Bayes Theorem
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2 votes

In my opinion, the events should be $X$: The number of white balls drawn from Urn 2; the three possible outcomes 2,1, and 0 define the partition you need. $B$: The ball drawn from Urn 1 is white; ...

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Proposing a stochastic model
2 votes

What you have there is a mixture of two exponential distributions. Write the density as $$f(x) = \frac{5}{6}\frac{5}{2}\mathrm e^{-5x/2} + \frac{1}{6}\frac{1}{3}\mathrm e^{-x/3}.$$ This describes a ...

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Batch Least squares
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2 votes

Each sample is generated according to $z_i^T H = y_i$. If you have $N$ data points (each one consisting of a three-dimensional vector $z_i$ and an observation $y_i$), you collect them in an $N\times 3$...

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Compute a probability given a Bayesian network using variable elimination
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2 votes

On this network, variable elimination will not result in a lot of simplification because of the connected structure (that is, not many variables will disappear when we do the elimination). We ...

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Compute probability given a Bayesian Network
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2 votes

You need to marginalize all the variables from the joint distribution: $$p(a,d,f,g) = \sum_{b,c}p(a,b,c,d,f,g) = \sum_{b,c} p(a)p(b|a)p(d|b)p(f|b,c)p(g|c).$$ You plug in $+d$, $+f$, and $\neg g$ and ...

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Calculate probability using brute-force method
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2 votes

The Bayesian network tells you that the joint distribution factorizes according to $$P(a,b,c,d,f,g) = P(a)P(b|a)P(c|a) P(d|b)P(f|b,c) P(g|c),$$ So you can compute any truth assignments using the ...

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argmin with logic statements [solved]
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2 votes

Judging from the text around your expression, it seems plausible that $T^\star$ is the smallest $T$ such that $\theta_T \geq E_0[T]$. These are not random variables because they are supposed to be ...

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Does replacing $\ge$ by $>$ lead to a strictly inequality too? (integral inequality)
2 votes

If the function in continuous, then you can use a lower-bounding argument as follows: If $f(x) > 0$, there is a small enough $\epsilon>0$ such that $f(x)>\epsilon$ for all $x\in [a,b]$, You ...

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