# Questions tagged [probability]

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26k views

### Define custom probability density function in Python

Is there a way, using some established Python package (e.g. SciPy) to define my own probability density function (without any prior data, just $f(x) = a x + b$), so I can then make calculations with ...
10k 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: ...
2k views

### Drawing samples from a finite mixture of normal distributions?

After some Bayesian update steps, I am left with a posterior distribution of the form of a mixture of normal distributions,$$\Pr(\theta| \text{data} ) = \sum_{i=1}^k w_i N(\mu_i, \sigma^2).$$ That is, ...
225 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. ...
2k views

### How can I generate half-normal variates in MATLAB?

I can find the random('normal', 0, 1, 10000,1) command in MATLAB but it generates half-normal variates. I would like to generate random half-normal variates. The ...
8k 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 ...
2k views

### How to sample numerically from an arbitrary smooth distribution?

I'm given a smooth probability density function via its values on a reasonable fine grid. I assume that cubic spline interpolation (or cubic spline interpolation of the logarithm of the density) will ...
637 views

### Efficient computation of Markov chain transition probability matrix

Consider a continuous Markov chain $X=(X_t)$ on a finite state space and let $Q$ be the (given) transition rate matrix. This matrix is very sparse, with non-zero values on 3 diagonals only (so from ...
902 views

### Computing the PDF of a quadratic function of two random variables

Given the function $\mathcal{M} = g + Ah + Bh^2$ where $A$ and $B$ are constants and $g$ and $h$ are random variables with their distributions $f_G(g)$ and $f_H(h)$ known, is it possible to compute ...
166 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 ...
176 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 ...
468 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 ...
376 views

### Convert ODE into discrete probabilistic model

how can I turn an ODE equation into a discrete probabilistic model? I take for example the Verhulst equation for the growth of a population. $$\frac{dP}{dt} = rP(1-P/K)$$ I was thinking to simulate ...
162 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 ...
148 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 ...
89 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. ...
48 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 ...
132 views

201 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 ...
129 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: ...
44 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 $x$ ...
322 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 ...
231 views

### Chinese Restaurant Process… Why?

I recently started to study non-parametric clustering methods and I come across to CRP. After reading all the material I found on the web there is one thing which is not completely clear to me: which ...
240 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 ...
61 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 ...
264 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 ...
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 ...
126 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-...
331 views

164 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: ...