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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21 views

normal quantile integration

Let $X$ be normal distributed with $\mu, \sigma$ and let $F$ and $f$ be the cdf/pdf. I would like to calculate the following integral: $I = \int_0^1 e^{(-ap)}F^{-1}(p)dp$ let $p = F(x)$, then $I = ...
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Choice probabilities and integration

I have a question from the book "Discrete Choice Methods with Simulation" (Train, 2009). In chapter 1, , part 1.2.3, partial simulation, partial closed form, it has the following equation which ...
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46 views

Why does integrating a probability density function give probability?

It is well know that integrating a probability density function gives probability, that is, $$P(X\geq a) = \int_a^\infty f_X(x)\, dx$$ where $X$ is a continuous random variable, $a$ is a scalar and $...
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Joint expectation subscript when we only have single random variable

As a follow-up to these two questions, Subscript notation in expectations and Double expectation (Not law of iterated expectation) If we have $E_{XY}[X]$, what is the correct integral expression? i....
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1answer
64 views

Calculating the sum of dependent uniform random variables

My question derives from Problem calculating joint and marginal distribution of two uniform distributions. So, suppose we have random variables $X_1$ distributed as $U[0,1]$ and $X_2$ distributed as ...
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30 views

Multi-dimensional Lebesgue integral applied to probability measures

I have a question on multi-dimensional Lebesgue integral when applied to probability measures. In particular, consider the random variables $Y,X,\epsilon,T,V$, with supports $\mathcal{Y},\mathcal{X},...
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32 views

Why do we integrate to obtain prior distributions?

In the test for difference in means with unknown variance, it's stated that in order to obtain prior distributions, we have to take the double integral with respect to mean and to the variance? ...
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23 views

Integrated time series and continuous time integral

I am new to time series, and have a theoretical (philosphical, if you wish) question on the connection of an integrated time series with the usual concept of integration in calculus. If $x_t$ is an ...
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27 views

Integration of (Phi(x)-Phi(y))^2d(F(x, y)

How to integrate the following? Thanks
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25 views

Integral and expected value for multivariate distribution

for the last couple of days I have been struggling with a problem and I was hoping to get some help here. I have a function that looks as follows: $$ f(x,y,z)=\begin{cases} 0 & \text{if} \ (x*...
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59 views

Why intuitively does $\mathbb E(\frac d {d\theta}\log p_\theta(x))=0$?

Let $p_\theta(x)$ be the probability density function of $x$. Then obviously, $\frac d{d\theta}\mathbb E(1)=0$. But note that $\mathbb E(1)=\int p_\theta(x)dx$, so that $\frac d{d\theta}\mathbb E(1)=\...
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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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54 views

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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88 views

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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59 views

Absolute moment and integration by parts

Let $X$ be a real continuous random variable with distribution $F$ with finite moments. I want to calculate $$E[\vert X \vert] = \int_{-\infty}^{\infty} \vert x\vert dF(x)= -\int_{-\infty}^{0} x dF(x)...
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24 views

Integrate $\int_{-\infty}^{\infty}\frac{1}{2\pi}e^{(-\frac{1}{2}(\frac{x^2}{4}+4y^2))} dy$

I'm trying to integrate $\int_{-\infty}^{\infty}\frac{1}{2\pi}e^{(-\frac{1}{2}(\frac{x^2}{4}+4y^2))} dy$ using the fact that the integral of any normal PDF is 1. But I'm having trouble completing the ...
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20 views

Phase marginal for a multivariate complex Gaussian density

Suppose $z$ is a random variable taking values in $\mathbb{C}^n$ and admitting the complex Gaussian density $p(z;W) \propto \exp{(-\frac{1}{2}z^*Wz)}$, where $W$ is Hermitian. Let $r$ be the vector of ...
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58 views

Integration by parts for multivariable cumulative distribution function

How can I integrate by parts $$\int_A (y_1+\cdots+y_n) \,dF(y_1,...,y_n),$$ where $F$ is the cumulative distribution function for some random vector, $A$ is some Borel bounded set in ${\mathbb R}^n$.? ...
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IQN: Confusion about distortion risk measure

In the paper "Implicit Quantile Networks for Distributional Reinforcement Learning", they define $$ \begin{align} Z_\tau&:=F_Z^{-1}(\tau)\tag 1\\ Q&:=\mathbb E_{\tau\sim U([0,1])}[Z_\tau]\tag ...
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1answer
405 views

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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77 views

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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206 views

Double integral involving the normal CDF

I need to compute (or best approximate?) the following integral $$\int_0^\infty \int_0^\infty (1 + \alpha u)^{-1}(1 + v)^{-1} \Phi\left(\frac{\beta}{\sqrt{\gamma + uv}}\right) \text{d}u \text{d}v,\...
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25 views

Why is this a valid step (expectation w.r.t posterior)?

Reading through this paper and on page 10 they use the step: $$\int q(\theta|D,\phi) \log p(Y|X,\theta) d\theta = E_q \log p(Y|X,\theta)$$ Now obviously I understand why they have written this as an ...
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How can I calculate $\int{f(\bar{x}|\theta)g(\theta)d\theta}$ when $\bar{X}$ and $g$ are both normally distributed? [duplicate]

I'm reading Berger & Sellke (1987) 1. On page 115, the following statements are given: $$ \begin{align} m_g(x) & = \int{f(x|\theta)g(\theta)d\theta} \\ g & \sim \mathcal{N}(\theta_0,\...
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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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1answer
47 views

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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63 views

$\mathbb{E}[\sigma(r)^2]$ 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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33 views

Moments of the Generalized Dirichlet distribution

I have been trying to solve the following integral. $ \int\theta_j \sum_{k=1}^K \theta_k \beta_{k,w} \prod_{k=1}^K \frac{\Gamma(\alpha_k + \beta_k)}{\Gamma(\alpha_k)\Gamma(\beta_k)} \theta_k^{\...
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1answer
82 views

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 ...
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114 views

weighted sum of posterior Dirichlet distributions

I have the following distribution: $q(\vec\theta) = \frac{\sum_k \alpha_k}{\sum_k \beta_k \alpha_k} (\sum_k\theta_k\beta_k) \frac{\Gamma(\sum_k \alpha_k)}{\prod_k\Gamma(\alpha_k)} \prod_k \theta_{k}^{...
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159 views

How to find integral's upper limit using optimize function without knowing the interval where it will occur

I'm working on the following to get the upper limit of my integration, as I know I need to have some idea where the point might occur but in real it can be any value between 0 and infinity. May I know ...
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1answer
43 views

How can I derive the function curve from a histogram of observed data

I'm analysing some datasets that produce heavy tailed data when plotted as a histogram. My initial goal was to attempt to fit a known distribution to my dataset. Thereafter I use to the properties of ...
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159 views

Limits of integration of a density function

My question is based on this post. In summary, $X \sim \text{Unif}(a,b)$ and $Y|X \sim \text{Unif}(a,X)$. Then the author does the following calculations: \begin{align} f(y) = \int_{-\infty}^{\infty} ...
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1answer
77 views

Marginal Distribution of Exponential Mixture Model

I am currently trying to marginalize over the scale parameter in a mixture distribution of exponential pdfs, but I do not trust my result. Let me show you my steps: Probability Density Function The ...
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150 views

Partial integration involving CDF [closed]

I am reading a textbook which claims that we can obtain by partial integration, for CDF $F(x)$:$$\int_{t}^{\infty} (1-F(x)) \frac{dx}{x}=\int_{t}^{\infty} (\log u -\log t) dF(u) $$ I am aware that ...
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297 views

How to calculate the integral of Normal CDF and Normal PDF?

I'm trying to find $\int_{\frac{a-b}{B}}^\infty\Phi\left(tA+ABx\right)\phi(x)\,dx$ where $A = \frac{\sqrt{\gamma_{3}+\sigma_3^2}}{\gamma_{3}},\ B = \frac{\gamma_{2}}{\sqrt{\gamma_{2}+\sigma_{2}^...
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1answer
39 views

How to evaluate $\int_0^\infty m^{x+1}e^{-2m}dm$ as $\Gamma(x+2)\frac{1}{2}^{x+2}$?

$\int_0^\infty \frac{m^{x+1}e^{-2m}}{\Gamma(x+1)\Gamma(2)}dm =\frac{\Gamma(x+2)\frac{1}{2}^{x+2}}{\Gamma(x+1)\Gamma(2)}$ How does the left side equal the right side? I understand that the gamma ...
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41 views

Piecewise integration

I am trying to estimate residential demand for electricity in a country where electricity is sold (to all households (HH)) at an increasing two-part tariff. By choosing marginal prices as my key ...
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1answer
109 views

Bayes version for continuous case, what does the integral mean?

In bayes version for continuous case, what does it mean to integrate with respect to $d\theta$ when $\theta$ is a vector not a a scalar value? $$p(\theta|D) = \frac{p(D|\theta)p(\theta)}{p(D)}$$ ...
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298 views

VAR model for first differences (not a good idea?)

I have read from couple of slides in the internet that if I have two $I(1)$ processes, it’s not a good idea to simply take the differences and include them in a VAR model, as then one might lose ...
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Integration of the product of Two multivariable Gaussain pdfs

I want to calculate \begin{align} \int_{-\infty}^{\infty} G(x-v_i, \Sigma_i) G(x-v_j, \Sigma_j) dx \end{align} where \begin{align} G(x-v_i, \Sigma_i) = \frac{1}{(2\pi)^{d/2} |\Sigma_i|^{1/2}} \...
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55 views

Moments of the horseshoe prior?

Are the first two moments well defined for the horseshoe prior? I would say that the expectation is zero but the variance does not exist. Using the following argument. Let $$\beta_i \mid \lambda_i, \...
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1answer
508 views

What is the difference between monte carlo integration and gibbs sampling?

I am aware that both are methods of sampling from the posterior. MC integration replaces the integral by a sample MC sample. Is this sample independent? Gibbs sampling is a class of MCMC ...
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1answer
2k views

KL divergence between which distributions could be infinity

I know that KL divergence measures difference between two probability distributions. My doubt is for which of the distributions it could become Infinity, putting it in another way, ...
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61 views

Convolution of a less typical distribution

$X_1$ and $X_2$ are independent and identically distributed (i.i.d) random variables defined on R+ each with pdf of the form $f_X(x) = \sqrt\frac{1}{2\pi x}exp[\frac{-x}{2}]\quad ,\quad x>0, \quad ...
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2answers
67 views

Calculating multivariate integrals between lower and upper bounds

Suppose $\vec{X}=(x_1,x_2,...,x_n)$ follows some continuous multivariate distribution, such that $x_i\in{\rm I\!R}, i=1,...,n$. Suppose also that I have access to the following functions: $\phi(\...
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1answer
238 views

Constant of Laplace approximation

I'm reading Example 3.16 of Robert & Casella's Monte Carlo Statistical Methods. It uses a Laplace approximation for approximating an integral related with the Gamma distribution namely $$\int_a^b\...
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36 views

Efficiently drawing from non-parametrically estimated distribution

Suppose I can estimate a distribution $G(x)$ as $$ \hat G(x) = f(x, X)$$ where $X$ are my data points and $f$ is a known, but computationally heavy function. Eventually I'm interested in $$ h(\...
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1answer
161 views

Relation of standard deviation between independent and dependent variables

Is there a relationship between the standard deviation of an independent variable to the standard deviation of a dependent variable? For example, if we know the standard deviation of a variable $x$ ...
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81 views

Kernels property: integral of kernel product $\propto k(x,y)$

Let $k$ be a kernel function (symmetric and semi-positive definite function). Does the following relationship hold: $\int_{-\infty}^{+\infty}k(x,u)k(y,u) du \propto k(x,y)$ ? Or for what type of ...