Questions tagged [probability]

For computation questions involving computing probabilities or simulating probabilistic processes.

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What are the Exact Rules for Significant Figures, Precision, and Uncertainty?

In the physical sciences (which are physics, chemistry, astronomy, materials science, etc.), we learned that the uncertainty is +/- the smallest unit (which is 1) of the last significant figure if the ...
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6 votes
1 answer
251 views

Estimate the number of self-avoiding walks of length $n$

For the past couple of days, I have been thinking a lot and searching online for an algorithm capable of estimating the number of Self-Avoiding Walks (SAW) of length $n$ in $\mathbb{Z}^2$. There is a ...
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2 votes
1 answer
137 views

Importance Sampling for multidimensional integrals and random numbers from multivariable pdf's

I am aiming to get a numerical value for a five-dimensional integral using Monte Carlo Integration. I am getting good results using the Mean Value Method, but I would like to try to use Importance ...
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-1 votes
1 answer
48 views

Generating Rvs for a given PDf in python

Two random variables $X$ and $Y$ are distributed according to \begin{align} f_{XY}(x,y)= \begin{cases} x+y & 0\leq x \leq 1, 0\leq y \leq 1 \\ 0 & otherwise \end{cases} \end{align} I ...
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Probability Density Function of DNS velocity field [duplicate]

I am currently working on a DNS turbulent solver and I would like to compare my IHT simulations to papers. Those papers show the Probability Density Function of the velocity field: I would like to ...
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1 vote
1 answer
107 views

Define continuous, non-analytical pdfs in python

I am planning to do some basic algebra on continuous, non-analytical random variabels. I want to define their probability density functions as arrays x and f(x). Yet, I was surprised to find out that ...
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N-dimensional optimization of moments of distribution function

Let's say I have a bag of marbles. Each marble has several attributes (color, diameter, surface roughness, weight). I know that there is a statistical relationship between various marble attributes ...
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2 answers
239 views

Different questions about "Inverse Physics problems"

I am in a context of forecasts in astrophysics. Don't be too rude if questions seem to you stupid or naive but rather indulgent, I am just looking for better undertsand all these numerical methods of ...
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5 votes
1 answer
275 views

Non-negative least squares with very small numbers

(I have asked this question on StackOverflow previously but it has been pointed to me that CSSE or MSE could be more appropriate) I have to solve a constrained optimization problem of the following ...
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32 views

Combining many probabilities, modifying, seeking general formula

CONTEXT I need to combine the probability of occurrence of many thousands of events for millions of individuals (trees) in an agent-based/individual-based simulation model developed in NetLogo (agent-...
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2 answers
279 views

How to solve system of equations with almost-zero determinant?

I have a system of equations that I am trying to solve. In matrix form, it's written as $$x(I - S) = b.$$ I am solving for $x$, where $I$ is the identity matrix and $S$ is a matrix where each column ...
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How to simulate a PDF using data samples like this?

I know two methods to simulate a PDF from random data samples using MATLAB : 1) Using a histogram where I use this command histogram(data,'Normalization','pdf'), it gives PDF like bins. 2) Another ...
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1 answer
69 views

Change of random variables and check by plot

Question As a test, I transform a uniform distribution over the unit square. But when I check the transformed distribution with Monte Carlo, it is wrong. What went wrong? Thanks. Problem Random ...
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110 views

difference between PLSA and LDA

What is the difference between Probabilistic Latent Semantic Analysis(pLSA) and Latent Dirichlet Allocation (LDA)?
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2 votes
1 answer
92 views

How to add extra constraints to a linear system for probabilities?

Background: I have an equation which looks like as follows: $W \times P = R$ $$\left[\begin{array} &{1}&{0}&{0}&-\frac{w_{1}}{w_{o1}} &\dots &{0} &-\frac{w_{1}}{w_{0} } \...
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2 votes
1 answer
257 views

Numerical integration of Fokker-Planck equation allowing for negative drift?

The Fokker-Planck equation (a.k.a Kolmogorov forward equation or Smoluchowski equation) describes the evolution of a probability density function and numerical integration of the FPE should conserve ...
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  • 123
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52 views

Uniformly sample a point per polytope

I want to uniformly sample a point within each of $10^5$ convex polytopes in each iteration of a solver. The polytopes in one iteration are completely different from the polytopes in another iteration....
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3 votes
1 answer
104 views

GPGPU/FPGA programming for Combinatorial Analysis

Recently, I have taken an interest in performing combinatorial analysis for the game of 21 (blackjack) and attempted to use my AMD APU to try and thread the program via the 4 cores on the chip. ...
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2 votes
1 answer
335 views

Finding probability vectors from an implicit equation

I have $q$ $n$-dimensional vectors $\vec y_i$ and a matrix $\hat B$ of shape $n\times m$. I'm looking for $q$ $m$-dimensional vectors $\vec x_i$ such that: $\vec y_i=\hat B \vec x_i$ each vector $\...
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3 votes
0 answers
51 views

Book Suggestion for Approximating Integrals using Random Partitions

Suppose I want to approximate the integral $\int_0^1 x^2\,dx$ using Riemann Sums or Darboux sums over random partitions of the interval $[0,1]$, Like in the image below: Here, A "random" partition of ...
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5 votes
1 answer
168 views

Kernel-based differentiation

Consider a $\mathcal{C}^1$ function $V:\Omega\rightarrow\mathbb{R}$ where $\Omega\subset\mathbb{R}^n$. If a random vector $X$ has a parametric density $p_\theta(\textbf{x})$ that's smooth in its ...
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  • 201
2 votes
1 answer
49 views

Verifying that ODE integration generates Theoretical Stationary distribution

I am trying to simulate an ODE, like $ \dot{x} = \xi(x) $ that should have a stationary distribution (a la Stat Mech). Assuming that my ODE algorithm generates time samples of my system state $ x $ ...
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  • 153
2 votes
0 answers
45 views

Deterministic method to compute "Process noise covariance matrix, Q" for a Kalman filter when parameter variations of the model is known apriori

I am implementing a Kalman filter (for a linear ODE system for now). My model represents a physical device that has 6 "parameters", i.e. those values of the device do not evolve over time (within a ...
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0 votes
1 answer
91 views

What is the expected value of area of intersection of a circle and a rectangle

$r$: cicle C1's radius $w$,$h$: rectangle R1's edges: $x=w$, $y=h$, $x=0$, $y=0$ $(w>2r, h>2r)$ $S(x,y)$: area of ...
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0 votes
2 answers
4k views

How to generate Poisson-distributed random numbers quickly and accurately?

I have attempted to create Poisson-distributed random numbers, seeing that it is not so easy as the simple multiplicative algorithm works accurately only if the mean is less than 500. Using logarithms ...
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  • 151
1 vote
1 answer
139 views

How is KDE used in stochastic tomography

I am currently writing my masters thesis and my topic also touches on Stochastic Tomography for volume reconstruction presented in this paper. Now i understand most of the process described, but i ...
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  • 113
3 votes
0 answers
140 views

Closed form PDF/CDF using Orthogonal Polynomial Expansion (gPC)

Consider a random variable which is given by an orthogonal polynomial expansion in one parameter (or polynomial chaos expansion PCE), i.e. , $$ f(\alpha) = \sum\limits_{n=0}^{\infty} \hat{f} (n) \psi ...
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1 vote
0 answers
112 views

Numerically compute PDF given a function

Consider $[0,1]$ with the Lebesgue measure $m$ and $f:[0,1]\to \mathbb{R}$, and $x$ a uniformly distributed random variable in $[0,1]$. Then, $f(x)$ itself define a new random variable. We can then ...
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2 votes
1 answer
68 views

Using physical parameter as a Gaussian random variable in a simple Poisson problem

I want to vary the input parameter of a physical dynamic mechanics problem, as a Gaussian Random variable and view the resulting Probability Density Function (PDF). I used the Finite Element Method to ...
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2 votes
1 answer
360 views

Optimize custom probability distribution in Python [closed]

Consider random variables $X$ and $Y$, their distributions are given. $Z = f_a(X, Y)$ where $f(\cdot, \cdot)$ is a deterministic, not random function $f_a: \mathbb{R}^2 \to \mathbb{R}$ depending on a ...
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2 votes
0 answers
445 views

Numerical methods for calculating the inverse CDF when closed form approximation not available

I need to calculate the inverse CDF for a probability distribution, however there is no closed form approximation available in the literature. The distribution I am working with is the Normal Inverse ...
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2 votes
1 answer
194 views

Probability of reconstructing a word using c substrings from a random sample

Consider a voice recording split into it's phonemes as our sample $S=(s_1,...,s_k) \in \Omega = P^k$. The number of phonemes is $|P| = 40$. Then I have a word $w = (w_1,...,w_n) \in P^n$. I want to ...
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5 votes
1 answer
634 views

Efficient and stable computation of inverse CDF

What is the most efficient and numerically stable algorithm for computing the inverse CDF $F^{-1}(y)$ of a probability function, assuming that both the PDF $f(x)$ and the CDF $F(x)$ are known ...
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7 votes
2 answers
229 views

Learning parameters of noise and filter coefficients from data where data and noise both have Gaussian distributions

Assume $X$ and $N$ are two sets of vectors (observations) from two different normal distributions, where $X$ represents clean data and $N$ represents noise; and $A$ a projection matrix of a filter. ...
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1 vote
0 answers
44 views

Probabilistic model to approach problem that is usually dealt with linear programming

I have the following problem. Let's say we have $x_{jk}$ it is an expression value of gene $j$ in a sample $k$. It is the average of expression levels across the cell types $s_{ij}$, weighted by ...
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0 votes
0 answers
20 views

Assigning new values based on original guesstimates and ranking / ordering?

Lets say we have two things as input, $N$ scalars (measurements) that we know are erroneous to some degree (i.e. the correct values are somewhat similar). In addition, we also have a roughly more ...
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  • 113
6 votes
1 answer
9k views

How do I generate Maxwell-Boltzmann variates using a uniform distribution random number generator?

I am doing a molecular dynamics simulation. I need to assign initial velocities to the atoms. I want to assign the initial velocities which follow the Maxwell-Boltzmann distribution. How do I ...
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1 vote
0 answers
178 views

Simple MCMC Algorithm in Matlab

I would be really glad to get some specific advise on how to implement a simple MCMC algorithm (in Matlab, if possible). I'm not yet too familiar with optimization methods. My problem goes as follows: ...
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2 votes
1 answer
273 views

Metropolis Monte Carlo integration of Area with unknown normalization

I probably miss something very basic. I don't see how to use Metropolis–Hastings algorithm for computation of integrals, if I don't know the volume of accessible phase space (i.e. proper normalization ...
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4 votes
1 answer
167 views

solving for unknown inside an expectation

I need to find roots for the following function: $$f(\theta) \equiv E[R(\theta;\eta)]=0$$ for some unknown $\theta$ which is deterministic, while the expectation is taken over a normally ...
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  • 235
3 votes
1 answer
152 views

Computing expectations

I want to compute the following conditional expectation $E_{t}[\phi(A_{t+1}, \eta_{t+1})| A_t]$ where $\log A_{t}=\rho \log A_{t-1} + e_{t}$ and $e_{t}$ is IID $N~(0,\sigma_e)$ and $\eta_{t}$ is ...
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  • 235
2 votes
2 answers
824 views

Python: What is a good way to generate a 1D particle field with a gaussian distribution?

If I have N particles how do I assign their x values so that the end result is Gaussian distribution. i.e. particles near the ends are more spread out than particles near the center.
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2 votes
2 answers
362 views

Extracting time scales information from empirical cumulative distribution function

I have a stochastic process (a Markov chain actually) that has two absorbing states. I am using a difference equation to calculate the first passage time to either of the absorbing states. There are ...
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  • 269
1 vote
1 answer
1k views

Coding complex equations into C++

I have here equations from a paper by E. Bradlow. They're for counting the events in a Weibull-distributed data set. $$\begin{align} \Pr(N(t)=n) &= \sum_{j=n}^\infty{\frac{(-1)^{j+n}(\lambda t^c)^...
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1 vote
2 answers
12k views

How to plot probability density function in MATLAB?

I'm trying to get a frequency plot, or PDF (probability density function) plot for my biometrics project in MATLAB. I have two vectors genuine_scores and ...
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  • 121
2 votes
2 answers
2k views

visualization of 3D probability flow

I have a master equation for $P(N_A^+,N_B^+,N_C^+,t)$, with $N_A^+,N_B^+,N_C^+$ all discrete. The numerical integration is done by this Matlab program using Euler's method. Despite the crude Euler's ...
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  • 269
2 votes
0 answers
32 views

Approximate convolution of independent Beta variates?

Is there a way to approximate the convolution of Beta variables? Specifically, I am trying to find an approximation to $g(x_0)$: $$g(x_0) = \int \delta(x_0-\sum_{i=1}^{n} a_i x_i) \prod_{i=1}^{n} f(\...
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2 votes
3 answers
134 views

Hash in set probability?

First off I would like to mention that I am absolutely terrible at statistics... so bear with me please. The question: a sha1 hash is a hexadecimal string of 40 characters, the largest number being: ...
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11 votes
2 answers
11k views

How do I plot the surface of a 4D plot?

I am trying to plot the wave function for a particle in a 3D box. This requires me to plot 4 variables: x, y, z axes and the probability density function. The probability density function is: ...
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  • 263
2 votes
1 answer
135 views

Multiple independent random number streams

Having multiple streams of pseudo-random numbers known to be independent and with a uniform distribution I want to do Monte Carlo simulations in parallel. In other words, one thread will have a full-...
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