# Questions tagged [numerical-integration]

A class of algorithms to approximate definite integrals.

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### Calculation of Posterior distribution numerically

For calculating posterior probabilities numerically, I did not understand that why is in the following codes they have divided by 0.001 in the denominator to calculate ...
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### Estimating the integral $\int_{0}^\infty x^4 e^{-2x}\,dx$

We have been given a random variable having a Gamma distribution as shown below: Using the accept-reject algorithm, we are supposed to sample from the Gamma distribution using exponential as the ...
191 views

### How to Find PDF of Transformed Random Variables Numerically?

Is there a way to compute and plot the transformed random variable in Python or R? Let's say if $X_1$ and $X_2$ are independent and identically distributed random variable with PDF defined over non-...
1 vote
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### Are discrete mixtures Gauss quadrature-like integral approximations?

I noticed that the formula for Gauss (or Newton-Cotes) quadrature looks very similar to the formula for the PDF of a general mixture distribution. Let $p_{comp}(x)$ be the PDF of a compound ...
173 views

### Why not just run a Markov chain to get stationary probabilities?

I'm reading Performance Modeling and Design of Computer Systems which contains some analysis of Markov chains. In particular, it emphasises various analytical methods for finding the stationary ...
917 views

### Estimating the cumulative probability of a bivariate normal distribution

I have a quick question regarding working out the probabilities of a bivariate normal distribution. To my knowledge, there is no nice closed-form for a cumulative distribution function for the ...
111 views

### Bayesian Quadrature to find expectation of unkown function w.r.t. known pdf

I am interested in estimating the integral $\int f(x) P(x) dx$, where $f(x)$ is an expensive function and $P(x)$ is has an analytic form. I would like to evaluate this with as few evaluations of $f(x)$...
108 views

### Approximate Posterior Predictive Quantiles with Numerical Methods

I have a posterior function which is easy to approximate using numerical methods (the posterior has only 2 parameters, and is approximately Gaussian because of the large sample). However, I need to ...
1 vote
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### What is the expectation of a random variable satisfying some conditions?

How to find the expectation E[X.I(Y<x,X<x)], where X and Y are independent random variables with respective cumulative distribution functions F(.) and G(.) respectively. x is a positive value. ...
926 views

### Plain English explanation of Ito's integral?

I'm looking for a plain English explanation of Ito's integral. I don't need an exhaustive proof, derivation, etc. Just a simple ~this is effectively what it does and why it's better than a Riemann sum ...
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### Expressing a marginal probability using copulas

Please correct me if I am wrong and kindly provide me with the correct notations. I have two questions: We know that for the variables $(X,Y,Z)\in \mathbb{R}^3$, the marginal joint density $f(x,y)$ ...
1 vote
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### How to calculate Integral error from MCMC

I've recently started using Markov Chain Monte Carlo to calculate integrals, but I can't seem to find how to calculate the error for such an integral. In the standard case of importance sampling, the ...
1 vote
369 views

### Computational Complexity of Multivariate Normal CDF

I'm going to post this here also as per user suggestions since I feel like the root cause of the issue is more maths related than code related. I'm working on a multivariate cross-entropy minimization ...
1 vote
139 views

### Finding confidence interval for unimodal function equivalent to and comparable with standard deviation of normal

I'm trying to characterise an arbitrary, unimodal distribution in a way that is a) easily understandable (to a physics audience) and b) comparable with a normal distribution. My thinking goes like ...
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### Why are my sampled values are non Gaussian?

I just have a quick question regarding Importance Sampling Monte Carlo integration. If I sample from some pdf, $p(x,y)$, to calculate an integral. I.e., $I = \int f(x,y) \ dx\ dy$ It can be ...
1 vote
645 views

### Integration with accept reject sampling Monte Carlo

I've got a quick question with regards to accept-reject Monte Carlo integration that I can't solve. Suppose I want to integrate some function, $f(x,y)$, with samples of $x, y$ from $p(x,y)$. Now, ...
152 views

### How to calculate individual moments of a 2-dimensional distribution via Monte Carlo Integration

I've recently been using Monte Carlo integration to calculate a particular integrals which I can do fine but I've hit a problem where I can't calculate the individual moments of my distribution for ...
1 vote
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### how do you find the vector that minimizes [closed]

How do you find the vector that minimizes $\|A\mathbf{x}-\mathbf{b}\|_2^2$ ?
I want to see the effect of parameter uncertainty in the Euler method for ODEs. For a differential equation: $dx/dt=f$ with initial condition $x(0)=xo$ and a function $f$ (that has uncertain ...