# Questions tagged [integral]

For on-topic question related to uses of the mathematical concept of an integral, i.e. $\int_a^b f(x)\; dx$. Purely mathematical questions about integrals are better asked at math SE: https://math.stackexchange.com/

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### Probability distribution function expressed in terms of a divergent series

I'm interested in finding the CDF and PDF of $U_i$ defined as follows, $$U_i=\frac g{d^{\alpha}}$$ where $g$ is a gamma distributed random variable with shape $k$ and scale $\theta$, and $d$ is a ...
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### Bayesian estimator $\theta(x)$

Given a training set of $(X, Y )$'s where the $X$'s are the source variables and the $Y$'s are the targets, derive an estimator that minimizes the mean squared error between target values and ...
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### How to calculate the integral?

When the function is , the second order differential is . How to calculate the expectation of the second order differential as below, where Finally, the result of expectation will be (L ...
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### Calculating the integral of the ratio of standard normal density and standard normal distribution functions

I'm trying to calculate the integral of $\int_{-\infty}^{\infty}\frac{\phi(ax+b)}{\Phi(ax+b)}\phi(x)\phi(ax+b)dx$ and find an analytic solution for it but with no luck. I tried integration by parts ...
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### when can integration and expectation be exchanged?

When is it possible to move expectations into integrals? In a proof of the Central Limit Theorem, at one point an expectation was moved into the integral (without much explanation of why that worked....
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### Integrating with a multivariate Gaussian

I need to figure out the steps to solve the following integral, where $Q(\mathbf{w})$ is a multivariate Gaussian with mean $\overline{\mathbf{w}}$ and covariance $\mathbf{C}$: \begin{align}\int Q(\...
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### Variance of Monte Carlo integration with importance sampling

I am following these lecture slides on Monte Carlo integration with importance sampling. I am just implementing a very simple example: $\int_{0}^{1} e^{x}dx$. For the importance sampling version, I ...
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### Solving integrals in R

I would like to write an R function for solving the following equation: Essentially I would like to be able to set or vary the parameters values of "m" and "s" and those parameters in "p(t)" ...
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### Integrating the inverse-Wishart density

It is alleged in this question and in the Wikipedia article and elsewhere that the density function for the inverse-Wishart distribution with $n$ degrees of freedom on $p\times p$ positive-definite ...
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### Expectation of $h \circ X$

I'm only starting to learn statistics. The definition I've been given for the expected value (expectation) of a continuous random variable X with probability density function (PDF) $f_X$ is the ...
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### $\mathbb{E}[\sigma(r)^2]$ with $r \sim \mathcal{N}(0,1)$

Start with a random variable $r \sim \mathcal{N}(0,1)$. Now consider the random variable $\sigma(r)$ formed by passing it through a standard logistic function $\sigma(x) = \frac{1}{1 + e^{-x}}$. I ...
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### Mean and variance of probability density with multidimensional indicator function

I encountered the following question while studying machine learning: We are asked to calculate mean and covariance of a given probability density function p(x) = \frac{1}{16} \cdot 1_{0 \leq x_1 ...