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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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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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1answer
22 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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10 views

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
316 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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0answers
39 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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2answers
190 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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1answer
24 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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21 views

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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42 views

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
45 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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0answers
58 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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24 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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40 views

Calculating statistics on transformations of many random variables [closed]

I have a model that uses $n$ independent random variables $X_1,..., X_n$. I know the density function of each random variable. I would like to calculate statistics such as $E(\sqrt{X_1+...+X_n})$ or $...
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1answer
60 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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1answer
72 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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117 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
41 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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1answer
157 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
61 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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108 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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1answer
198 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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22 views

How to integrate probability density of sum of two indepedent random variables with a finite lower bound on one of them?

$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-u^2/2}\:du=1$$ but $$u = \ln(A)-C-k$$ where $\ln(A)$ and $C$ are normally distributed independent random variables, and $k$ is a constant. I am ...
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35 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
98 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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1answer
127 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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0answers
21 views

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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0answers
48 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
272 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
629 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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2answers
58 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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Survival probability in economics

I try to deal with survival probability in economics. Hope the question fits the site. I am trying to integrate by parts by using an indicator function. However, I am not really sure if it is a ...
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2answers
51 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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19 views

Back transform the integral of a log-log model?

One of my colleagues has an issue with back transforming the integral of model back to its original units. His model has a log transformed Y as a function of a log transformed X predictor. He can ...
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1answer
229 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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2answers
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
58 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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68 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 ...
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1answer
43 views

Find the value of an integral using Monte-Carlo method

I have a task in a subject called "Monte-Carlo methods" with which I'm a bit stuck and therefore I'm asking for your help. The task is as follows: Describe in detail one specific option how to find ...
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1answer
49 views

Mass point delta and it's mathematical interpretation

Considering a spike-and-slab prior of the form $$w\sim\pi\mathcal{N}(0,\alpha^{-1})+(1-\pi)\delta_0$$ where $\delta_0$ is a point mass at zero, if we would like to integrate over w such that $$I=\...
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Conditional Expected Value of Birnbaum-Saunders Bivariate

Consider $(T{1},T_{2})\sim BVBS(\alpha_{1},\beta_{1},\alpha_{2},\beta_{2}).$ According to item (b) of Theorem 3.1 of Balakrishnan and Kundu (2010) article, we have: \begin{align*} f_{T_{1}|T_{2}=t_{2}...
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58 views

Search solution to an integral

When I try to solve a problem of a personal research, I get the following integral: $$E\left(X\sqrt{(\alpha X)^{2}+4}\right)=\displaystyle{\int\limits_{-\infty}^{+\infty}\frac{x\sqrt{(\alpha x)^{2}+4}...
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194 views

Calculating Jensen-Shannon and Kullback-Leibler for density splines

I need to find a way to calculate the JS distance (and, by necessity, the KL divergence) between continuous, non-normal (generating distribution unknown) distributions. I have many classes, and ...
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1answer
57 views

Simulation of integral to infinity

I need to estimate an integral that has the next form: $$\frac{1}{\epsilon}\int_{q_\alpha}^\infty S(x)dx$$ where $S(x)=1-F_S(x)$, $\epsilon$ is close to $1$. The value $q_{\alpha}$ is a quantile at ...
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56 views

Stationary density of $\{X_t\}$ as a solution of integral equation

For the model, $X_t = \alpha X_{t-1} + \epsilon_t$, we find the integral equation related to stationary distribution in the following way: Let $X_{t-1}\thicksim f$ and $X_t|X_{t-1}=x \thicksim q(y|x)$...
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1answer
1k views

Finding the slope at different points in a sigmoid curve

This is my data. x <- c(0.5,3.0,22.2,46.0,77.3,97.0,98.9,100.0) plot(x, pch = 19) I want to fit a curve through these points and then calculate the slope at ...
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2answers
145 views

Expected value of $X$ which follows a normal distribution, between a certain interval [duplicate]

What is the process of finding the expected value of $X$ in a normal distribution between a certain interval? In particular I want to find: $E(X | a \le X \le b)$. For example, if $X$ has $\mu=0$ ...
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1answer
39 views

What is the expected value of X conditional on Lambda where both X and Lambda are random variables

Suppose we have a random variable $X \sim Expo(\lambda)$ with support $X \in \mathbb{R^+}$. $$f_X(x) =\lambda e^{-\lambda x}$$ Let us now also assume that $\lambda$ is itself a random variable and ...
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1answer
191 views

Standard Deviation of Cauchy distribution on a given interval

In general Cauchy distribution doesn't have standard deviation defined, though it should be possible to calculate it for a given interval. This is the formula that I'm trying to use: PDF for Cauchy ...
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2answers
33 views

Conditional independece iff joint factorize four variables

Im interested with a derivation as shown by by Zoubin Ghahramani in his article ' Learning Dynamic Bayesian Network' The whole objective was to prove P(Z, W|X,Y) = P(W|Y)P(Z|X,Y) ---- EQ 1 Given ...