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-3
votes
0answers
15 views

How to calculate quantization step of signal? [on hold]

Given the problem: Define the quantization step of a signal x(t) = 2*t + 1 on interval [3,7]. The number of quantization levels is equal to 32. A) 0.5 B) 0.20 C) 1 D) 0.12 E) 0.25
1
vote
0answers
37 views

2-step sampling from a conditional density

The setting is as follows: We are given two random variables $X : \Omega \to \mathbb{R}$ and $\Theta : \Omega \to T$ for some 'parameter space' $T \subset \mathbb{R}$, and 1) we know the density of ...
3
votes
2answers
172 views

Sampling Theorem for non-bandlimited Functions

The classical Shannon sampling theorem states that a bandlimited function with $\mbox{supp } \hat f\subset [-1/2,1/2]$ can be uniquely determined by its samples $(f(i))_{i\in \mathbb{Z}}$ (The symbol ...
1
vote
0answers
52 views

Simulate a graph from a certain distribution

I am wondering if anyone can indicate whether the following is a solved problem. I don't care about time of the algorithm currently. Consider a general probability distribution F on simple graphs ...
5
votes
1answer
90 views

Sample integer points of cross-polytope uniformly

For $r,d\in\mathbb{N}$, let $$C_{r,d}=\{x\in\mathbb{Z}^d: \|x\|_1\le r\}\subset\mathbb{Z}^d$$ be the set of integer points of the $d$-dimensional cross-polytope with radius $r$. What is ...
0
votes
0answers
36 views

Supremum distribution of band-limited functions with random spectrum

Consider the properties of band-limited functions $f_N:[-\pi,\pi]\to\mathbb{R}$ defined through their Fourier series $f_N(x)=\sum_{n=-N}^N c_n e^{inx}$ where $c_n=a_n+i b_n$ and both ...
0
votes
0answers
26 views

Efficient sampling from a polytope with large number of contraints [duplicate]

As far as I know, the most popular way to sample from a polytope (in H-representation) \begin{equation} \mathcal{P} := \{z \in \mathbb{R}^n | (Az)_j \le b_j\; \forall j=1,2,\ldots,m\} \end{equation} ...
5
votes
1answer
170 views

Exactly sampling from a distribution with access to the probabilities only

There is a discrete distribution where integers, $k$, from $1$ to $n$ occur with probability $p_{k}$, all $p_{k}$ are unknown. Rather than having access to the distribution we have access to $n$ ...
1
vote
1answer
216 views

Discretizing a cosine function?

I'd like to start by noting that for some fixed natural $N$ basis functions for my system will be generated by $f(k,x)$ as defined and explained here or in numerous other sources: $$f(k,x) = \sqrt2 ...
1
vote
3answers
287 views

A sampling and learning question

Suppose there is an oracle that returns a number $b \in \mathbb{Z}_{n}$ whenever I press the button. We have $b = a + e$, where $a \in \mathbb{Z}_n$ is a fixed number and $e$ is sampled according to ...
8
votes
4answers
139 views

Uniform Sampling Subject to Linear Equalities and Non-Negativity Constraint

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints $$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq ...
3
votes
2answers
354 views

Approximating Probability Distribution by Sampling

Consider a discrete probability distribution over $n$ events. Assume that the probabilistic kernel is a black box, that is, we can only sample from it without knowing anything about the type or ...
2
votes
0answers
125 views

Probabilities involving Beurling density

I am interested in calculating probabilities involving Beurling densities. Since it's likely probabilists are not familiar with the definitions, I give them below. Definitions. A metric space is ...
2
votes
1answer
242 views

Sampling point uniformly at random satisfying equality constraints

First of all, I apologize in advance if the question has already been asked in some way on this site and/or if there is a widely known solution to this problem. The description of my problem is ...
2
votes
1answer
78 views

Estimating mean and variance of a distribution based on error-prone estimates of its cdf

Suppose I have some random variable $X$ taking values in $[a, b]$ with unknown distribution (I am happy to assume the distribution is smooth, though it would be nice to not have to). I have a ...
0
votes
1answer
127 views

Monte Carlo estimator with autocorrelated samples

Given an integration problem $I=\int{f(x)dx}$, we can construct an ordinary Monte Carlo estimator as $E[I]=\sum\limits_i\frac{f(x_i)}{p(x_i)}$ where the samples $x_i$ are usually i.i.d. and drawn ...
6
votes
0answers
153 views

Sampling from a Convex Body with Many Extremal Points

Let $p_{1}, \ldots, p_{N}$ be a collection of points in $\mathbb{R}^{n}$. I would like to sample uniformly from the convex hull of these $N$ points in an `efficienct' way. In my setting, I have $n$ ...
3
votes
1answer
407 views

When can you describe a population and its component subpopulations with the same parametric family of distributions?

I believe that it is often the case that you are trying to select the best probability distribution to use to describe some phenomenon you are studying, and you have data not only for a population, ...
1
vote
2answers
160 views

Sampling uniformly from all possible line segments of a given length that fit inside a container

Consider that task of randomly placing a line segment of some length $L$ near a plane s.t. a point $p$ at the center of the line segment is at most a distance $H$ from the plane and intersections ...
1
vote
0answers
67 views

Stochastic process inference from partial observations

Consider a set $U$. My signal is a piece-wise constant "function" $Sig: t \mapsto s$, i.e. the signal at time $t$ equals to some subset $s \subset U$. One can see $Sig(t)$ as a stochastic process. ...
0
votes
1answer
431 views

transform a polynomial into another one upto a constant

I have a polynomial $p(x)=a_Nx^N+a_{N-1}x^{N-1}+\dots+a_0$. I want to convert this into another polynomial of same order, say $b_Ny^N+b_{N-1}y^{N-1}+\dots+b_0$. Is it possible to find a transformation ...
2
votes
2answers
624 views

Sampling without replacement until hitting a subset

I randomly sample uniformly from $ \{1,..,N \}$ without replacement until drawing a number $ \leq k$. Denote the expected number of draws by $R(N,k)$. I want a good approximation for $\sum_{k=1}^N ...
0
votes
2answers
440 views

Estimating a sum

Sorry for the vague title but I couldn't find a better one. I want to compute the sum $S = \frac{1}{N}\sum_{i=1}^N c_i x_i$ where $c_i$s are known positive constants. The problem is that computing ...
0
votes
1answer
2k views

Uniform sampling hemisphere and project in a specific direction [closed]

Hi, I need to generate a 'uniform sample' over an hemisphere and once done project it in a specific vector direction. I have try the following, but it produce some errors... maybe you have an idea ? ...
4
votes
1answer
271 views

Is it possible to sample uniformly on the surface of a high-dimensional polytope?

There are some pretty simple methods to do uniform sampling on the surface of high-dimensional spheres or cubes. Are there any methods that sample uniformly on the surface of a high-dimensional ...
6
votes
2answers
1k views

Sampling uniformly from a sphere

Let $B^{n} _p= ${$ (x_1, \dots, x_n) : |x_1|^p + \dots |x_n|^p = 1 $} be the unit ball in $\mathbb{R}^n$ in the $\ell^p$ norm. If $X_1,\dots,X_n$ are iid $\exp(1)$ -distributed random variables, then ...
6
votes
3answers
2k views

Random Sampling a linearly constrained region in n-dimensions…

Hi, So here is my problem: Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$ $x_n \le c_n$ and $\sum_{n=1}^N x_n = 1$ find an ...
2
votes
0answers
244 views

Sampling from a partition of a hypercube by convex polytopes.

I have a binary space partitioning (BSP) tree which recursively partitions a hypercube using linear hyperplanes. That is, a hyperplane splits the hybercube in half, creating two convex polytopes. Each ...
14
votes
2answers
881 views

Sampling from the Birkhoff polytope

The set of $n\times n$ real, nonnegative matrices whose rows and columns sum to one forms the well-known Birkhoff polytope Recently someone asked me if I knew How to sample (in polynomial time) ...