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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41 views

Closed form of the integral of the difference of two Gaussian CDFs?

Problem I'm trying to find the simplest form of the difference of two Guassian CDFs, i.e. $$ \int_{-\infty}^\infty \left( \Phi\left(ax+b \right) - \Phi\left(cx+d \right) \right) dx $$ for $\Phi(...
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22 views

What is the analog of the PDF and CDF for the likelihood function?

In probability, we can find the cdf using the pdf and vise-versa. Integrating pdf yields the cdf. Does integrating the likelihood function yield any important thing? In statistics, $\mathcal{L} (M\...
4
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1answer
135 views

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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1answer
52 views

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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31 views

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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0answers
45 views

Derivation of equation for Kendall's $\tau$ for a Frank copula

I am trying to prove the equation for Kendall's tau given in Nelsen (1986). In Section 3 , Equation 3.1 is a general form for any copula. Under the case of Frank, $\frac{\partial}{\partial u}C_\...
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18 views

Is the following identity regarding expectation of a normal random vector correct?

Let $\mathbf{X}\sim \mathcal{N}(\mathbf{0},\mathbf{I}_n)$. Let $g:\mathbb{R}^n\to \mathbb{R}$ be a continuous function. Then is the following identity correct? $$\mathbb{E}_{\mathbf{X}}\left[\delta(...
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1answer
34 views

Parameter in expectation subscript

There are a few questions/answers out there about subscripts involving a random variable ($E_X[...]$) or a density ($E_{f(X)}[...]$) in expectations. I like this one. But today I ran into a ...
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1answer
157 views

What does “the denominator does not contain any theta dependence” mean in Bayes' Rule? [duplicate]

Every lecture and book says that the denominator in Bayes' Rule does not depend on the parameter $\theta$. However, the denominator also includes $\theta$ in the formula of Bayes' Rule. I just cannot ...
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14 views

Distribution of squared Brownian Motion Conditional on its integral

I am interested in the distribution of $W_1^2$ conditional on $\int_0^1 W^2$. Simulations suggest the conditional mean is $\int_0^1 W^2-1/2$, and the variance is approximately one half that, so it ...
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2answers
55 views

Evaluate an integral using importance sampling

Estimate $\int^{1}_{0}e^{x} dx$ using importance sampling. Should I use beta distribution as proposal distribution and uniform distribution as target ?
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1answer
31 views

Symmetry between integrals including absolute value

So I came across below symmetry in my probability course that I can't understand. I understand how the lower bound changes when removing the absolute value operator, but how does the 2 disappear?
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47 views

Closed form solution for integral involving inverse Mills ratio?

Is there a closed form solution for the integral $$\int_0^a \sqrt{c} \phi\left(\frac{\mu}{\sigma}\sqrt{c}\right) m e^{-mc} dc$$ where $\phi(x)$ is the standard normal pdf? If it exists what is it? I'...
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1answer
46 views

Find conditional pdf given joint

Let the joint pdf of $X$ and $Y$ be $f(x,y) = 12e^{-4x-3y}, x>0, y>0$. What is the marginal cdf of $X$? of $Y$? Am I just supposed to integrate f(x,y) with respect to $x$ or $y$ to get the ...
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15 views

How to implement single Imputation from conditional distribution?

In [*] page 264, a method of drawing a missing value from a conditional distribution P(X_mis|X_obs;Theta) which is defined as: I did not find any code ...
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1answer
41 views

Understanding Gauss-Hermite Weights

I routinely use Gauss-Hermite as a tool for approximating complex integrals. While I am proficient in its applications, I am not proficient in its development. I am working to understand the weights ...
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4answers
3k views

Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?

For a uniformly distributed variable between 0 and 1 generated using rand(1,10000) this returns 10,000 random numbers between 0 and 1. If you take the mean, it ...
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0answers
20 views

How to obtain value range of empirical PDF, given the mode and area?

Given an empirical PDF of a continuous random variable $X$, then integrating over its entire defined domain will yield an area of size 1. To find the probability of $X \ge x_1 \land X \le x_2 $ (as in ...
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1answer
29 views

On continuous random variables

Mia breaks glasses at the rate of 4 per week .Let t be the time in weeks between successive breakages of glasses .Then: $f(t)= 4e^{-4t} \quad \text{when} \quad t\geq0 $ $ 0 \quad \quad \quad \...
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1answer
52 views

How to compute $E[ (|X|) X]$ when $X \sim N(0,1)$?

Any help would be appreciated.
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24 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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17 views

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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1answer
76 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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17 views

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
175 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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0answers
31 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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33 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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0answers
37 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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29 views

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

How to integrate the following? Thanks
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28 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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1answer
63 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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22 views

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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1answer
494 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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2answers
118 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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1answer
79 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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1answer
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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0answers
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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1answer
68 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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0answers
13 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 ...
4
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1answer
439 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
151 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
232 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
27 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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24 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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0answers
54 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 ...
2
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1answer
53 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 ...
3
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0answers
67 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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1answer
92 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 ...
3
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1answer
128 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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0answers
186 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 ...