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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51 views

Quasi-likelihood function

I got stuck in the derivation of the quasi-likelihood function. Namely, given an i.i.d sample $\{Y_i,X_i \}_{i=1}^n$ with $n$ the sample size, let the conditional mean and variance functions be ...
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1answer
113 views

Integration of a equation [closed]

$$\int_{x}^{y}\left[\sum_{i=1}^{N}\sqrt{a}\cos\left(\frac{2\pi(d_{i}-a)}{\lambda} \right)\right]^{\!2}da$$ Can anyone solve this integration for me I don't know how the summation and integration will ...
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0answers
17 views

Finding the marginal distribution of log-normal random variable whose mean is dependent on a Gaussian random variable

My goal is to be able to integrate out the observation error of $\hat{X}$ in the set-up below, in order to compute the likelihood of $\frac{X}{\hat{X}}$ over all possible values of $\hat{X}$ : The ...
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0answers
38 views

Derive Gini coefficient of lognormal distribution from definition

The Gini coefficient of a lognormal distribution $\operatorname{Lognormal}(\mu, \sigma^2)$ is $\operatorname{erf}(\sigma / 2)$, where $\operatorname{erf}$ is the error function. But how do I derive ...
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0answers
48 views

E-Step in EM algorithm with multiple latent variables

Within EM, we conduct the E-step in order to marginalise out parameters which we can view as ‘missing’ to then find easier modal estimates of parameters of interest. Suppose the parameters to be ...
2
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1answer
33 views

Expectation of the Log. of the Survival Function

I am doing some computations for a project and at some point the following expectation shows up $$ E_{x}[\log(1-F(x))] = \int \log(1-F(x)) f(x)dx $$ i.e the expectation of the natural logarithm of ...
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0answers
99 views

Deriving the risk of the Hodges-Le Cam estimator under squared-error loss

In order to better understand the behaviour of the Hodges-Le Cam estimator, $\tilde{\theta}_n$, I am trying to derive an expression for the risk $R_n(\tilde{\theta}_n, \theta)$ under squared error ...
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0answers
17 views

Integrating Monte Carlo Error Out of Likelihood Function

I am calculating the likelihood for a multivariate normal process in which the conditional mean in computed with Monte Carlo integration. I'm trying to account for the Monte Carlo error within the ...
2
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2answers
60 views

Express expectation value of a joint distribution over a discrete and continuous random variable

Let $Y$ be a discrete random variable and let $X$ be an (absolutely) continuous random variable and $f(X, Y)$ a function of these two random variables. Let $P(X, Y)$ be the joint probability measure. ...
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0answers
103 views

The expected value of log Gamma function

Suppose $X$ is exponentially distributed with the rate parameter $\lambda$. If we have the expected value of $\log X$ as \begin{equation} \langle \log X\rangle=-\gamma-\log\lambda \end{equation} where ...
2
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1answer
91 views

Order Statistics: How to calculate expected value of a function involving first and second order statistics

I am currently stuck with a challenging problem. I have n values drawn i.i.d. from a distribution F(x). Let $v_1$ be the nth order statistic (highest value) and let $v_2$ be the n-1 order statistic (...
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15 views

Alternative of the Nyström method

Say you want to obtain the eigenvalues/vectors of the integral operator associated to a kernel $K$. I know there is the Nyström method to obtain an approximation of these eigenvalues/vectors. What are ...
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56 views

What is $E(e^{rX^2})$ when $X~N(0,\sigma^2)$? [duplicate]

This is part of a homework problem. I think I've got it right, but it isn't fitting into the next step, so wanted to check this one. Question: What is $E(e^{rX^2})$ when $X$ is $N(0,\sigma^2)$? My ...
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7 views

Predictive distribution of multinomial probability with random effects

Let us say that we are trying to guess the correct category $c_i$ (there are $j=1,\ldots,J$ cateogries) for some items $i=1,\ldots,I$. Let's assume a multinomial regression model (aka softmax ...
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0answers
10 views

Area under Gaussian Process Regression for an interval?

For no practical reason whatsoever, simply curious: Say you need to use Gaussian Process Regression to model some $x,y$ relationship. Now, suppose that you're interested in integrating over some ...
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0answers
19 views

Help with computing convolution of gaussian and dirac delta

I'm trying to calculate message passing in Trueskill factor, Trueskill paper. Given only two players competing, the message from difference factor to winner team node t1 would be $$ \begin{align} m_{...
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0answers
77 views

Rewriting integral/summation as weighting estimator

I recently read a biostats paper which featured the following identity: $$ \sum_{y, l, m} y P(y, l, m \mid c, a) \frac{P(l \mid a, c) P\left(m \mid a^{*}, c\right)}{P(l, m \mid c, a)}=E\left(Y \frac{P\...
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1answer
242 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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2answers
82 views

Marginal distributions given the distribution of range

I'm working with an upper diagonal distribution whose distance from the diagonal is Lomax Pareto (Type II) distribution. The distance of a point from the diagonal line y = x is $\frac{\sqrt{(x_0-y_0)^...
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0answers
25 views

How can integral be interpreted as linear model?

I've searched how can I apply linear transformation to following problem, but it seems like there's just not intuitive explanation enough for me. We have $$ p\left(\mathbf{v}\mid \mathbf{h}\right) = \...
2
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1answer
36 views

Interpretation of integral

Does the integral below have a specific interpretation in statistics? It looks like the marginal expectation of y, without integrating over x. $$ \int_{y=-\infty}^{y=\infty} y f(x,y) dy$$
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1answer
422 views

Limit of Integration of continuous function

How to evaluate the following limit- $$\lim_{n \to \infty} \int_0^1 \int_0^1\cdots\int_0^1 f \bigg(\frac{x_1 + x_2 + \cdots + x_n}{n} \bigg) dx_1 dx_2....dx_n$$. Here $f()$ is a continuous function $f:...
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1answer
99 views

Calculation of integrals transforming $N(μ,σ^2)$ to $N(0,1)$

Let's say $X\sim N(\mu,\sigma^2)$, where $\mu$ and $\sigma^2$ are known. How can we calcuate the following integrals by transforming $X$ to $Z\sim N(0,1)$? $$ \int_{c_1}^{c_2}(x-c)\frac{1}{\sigma\sqrt{...
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0answers
32 views

Marginal likelihood of bivariate Gaussian model

I assume the following model for a sample $y_1 \in \mathbb{R}^2$ of size $1$ with bivariate Gaussian likelihood and independent bivariate Gaussian and inverse-Wishart prior for the mean and variance ...
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0answers
13 views

Optimising Discriminator for GAN

For Generative Adversarial Networks, when trying to optimise for a Discriminator given a fixed Generator, we arrive at the following cost function (as in the original paper) $V(G,D)=\int_{x}p_{data(x)}...
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7 views

Marginal Distributions obtained by restricting a 2D Gaussian to a circle

Suppose I have a 2D Gaussian $$ f(x, y) = \frac{1}{2\pi\,\sqrt{\text{det}(\Sigma)}}\exp\left\{-\frac{1}{2}(\boldsymbol{x}- \boldsymbol{\mu})^\top \Sigma^{-1} (\boldsymbol{x}- \boldsymbol{\mu})\right\} ...
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0answers
13 views

Bayes Decision and Bayes Error, Probabilistic pattern recognition

Consider a classification problem where $X = (X1, X2) \in R2$ is uniform in $[- 1, 1] × [- 1, 1]$, and the conditional probability of Y = 1 is given by $\eta(x^1,x^2) = \frac{1}{4}(x^1 + 1)(x^2 + 1)$. ...
2
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0answers
48 views

multivariate gaussian integral

I am looking for the solution of multivariate gaussian integral over a vector x with an arbitrary vector a as upper limit and minus infinity as lower limit. The dimension of the vectors x and a are p $...
2
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1answer
38 views

About transformation of Variables

In a probability chapter of a Python Book, there is the following problem involving a transformation of variables: I don't fully understand where the value 1/z+1 in Y > X(1/z+1) comes from, and ...
0
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1answer
26 views

Expectation of ratios of probability density functions

I'm trying to solve/simply the expression below:- $\large \mathbb{E_{x \sim b(x)}} B\ [log\left(1 - \frac{A\ a(x)}{2\ c(x)}\right)]$, or $B \large \int_{x}b(x)log\left(1 - \frac{A\ a(x)}{2\ c(x)}\...
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0answers
29 views

High-dimensional integrals with computationally tractable closed-form solutions

What are some high-dimensional integrals with multimodal integrand and correlated variables, that also have computationally tractable closed-form solutions? The goal of the above constraints is to ...
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0answers
25 views

Derivation of an Equation by Owen

How did Owen derive this relation? in terms of normal cdf G(x), the normal pdf G'(x) and Owen's T-function. I cannot find the derivation nor the derivative of the T-function (only its definition both ...
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0answers
38 views

Analytical approximation of $\int_{-\infty}^{+\infty} l(x)~\mathbb{1}_{x\geq a}~\mathbb{1}_{x\leq b}~~\mathrm{d}x$ [closed]

I have an integral where $a,b\in\mathbb{R}$ but their order is not known and I want to compute $$\int_{-\infty}^{+\infty} l(x)~\mathbb{1}_{x\geq a}~\mathbb{1}_{x\leq b}~~\mathrm{d}x = \mathbb{1}_{b \...
3
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0answers
36 views

Integral of difference of density functions of two Continuous Random Variables goes to 0

The problem says : Let $(X_n)_{n=1}^\infty$ be a sequence of continuous random variables with probability density functions $(f_n)_{n=1}^\infty$ , and let $X$ be another continuous random variable ...
1
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0answers
45 views

How do I evaluate this Integral?

I’m reading Glen Cowan’s book on statistical data analysis and was stuck on an integral that has to do with crv transformations. I’m not able to evaluate the integral $$ g(a)da = \int_{x(a)}^{x(a)+|\...
0
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1answer
35 views

Integration with respect to the variance of a normal random variable

Let $X$ be a standard normal random variable and $f$ a measureable function on $[a, b]$. For $t \in [a, b]$, we know that $f(t) X$ has distribution $N(0, f(t)^2)$. Is it true that $\int_a^b f(t)X dt$ ...
1
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1answer
22 views

positive part of a random variable written as an integral

It is known that a random variable $X$ can be written in terms of positive and negative parts $X=X^+-X^-$. Also, $X^+=\max(0,X)=\int_0^\infty \mathbb I(X>x)dx$. Do you know how to show the equality ...
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0answers
20 views

In real-life applications, which continuous distributions have NON-CONVERGENT expectations that require Lebesgue integration?

When computing expected values, Riemann integration works for only random variables with bounded support sets. For distributions with unbounded support sets, we can use improper Riemann integrals for &...
3
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3answers
276 views

Expected value of a bivariate distribution as an integral

Let's assume an absolutely continuous random variable, $X$, with PDF $f(x)$. $$\mathbb{E}\big[X\big] = \int_{\mathbb{R}}xf(x)dx$$ If $X\sim f(x_1,x_2)$ is multivariate, then it makes sense to me to ...
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1answer
35 views

Marginal Distribution for x [closed]

I want to calculate the marginal distribution of $X$ given that the joint probability density function of $(X,Y)$ is given by $$f(x,y)=2592(x^2-y^2)e^{-2x} \qquad 0<x<\infty,\ -x<y<x$$ My ...
1
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1answer
65 views

Proving the predictive distribution in “Pattern Recognition and Machine Learning” by Bishop

I'm studying this book as it is the teaching material for one of the courses I'm taking now. In page 31, it says the predictive distribution is given by the integral of the likelihood function times ...
6
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1answer
64 views

Which $\mu$ hold so that integral of CDF (from $\mu$ to $\infty$) equals to integral of 1-CDF (from $-\infty$ to $\mu$)?

What is the $\mu$ s.t. $$\int_{\mu}^{\infty}1-F(x)dx = \int_{-\infty}^{\mu}F(x)dx?$$ Here $F(x) = P(X\leq x).$ Should $\mu$ be the median of X, i.e. $0.5=F(\mu)$? I think $\mu$ should be the point so ...
0
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1answer
49 views

Student-t Fisher Information wrt log(sigma)?

I am having some issues finding any information on how to compute the fisher information of a student-t distribution when the standard deviation is parameterized as $log(\sigma)$ rather than $\sigma$. ...
1
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1answer
91 views

Help with an application of Leibnitz's Rule

I'm trying hard to understand and solve the following: $$f_Y(y)=\frac{d}{dy}F_Y(y)=\frac{d}{dy}\int_{-\sqrt{y}}^{\sqrt{y}}{f_X(x)}dx=?$$ The background information is that $f_X(x)$ is the pdf of ...
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0answers
29 views

Why $E[E[p_{N}(Z_n,Z_m)|Z_m]]\neq E[E[p_{N}(Z_n,Z_m)|Z_n]]$ here? What went wrong?

We have a U-statistic defined as $U=\frac{1}{N(N-1)}\sum_{n\neq m}p_{N}(Z_n,Z_m)$, where observations $\{Z_i\}_{i=1}^{N}$ are i.i.d. following distribution $F(z)$ with density $f(z)$. $p_{N}(Z_n,Z_m)=\...
2
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0answers
28 views

Z = X1/(X1+X2) where X1 and X2 are gamma distributed [duplicate]

Suppose that $X_1 \sim \Gamma(\alpha_1,\beta)$ and $X_2 \sim \Gamma(\alpha_2,\beta)$ and let $Z = \frac{X_1}{X_1 + X_2}$ ($X_1$ and $X_2$ are assumed to be independent). I want to prove that $Z$ is ...
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0answers
150 views

How to prove that the convexity of Generalized Linear Models?

Refer to question: Does log likelihood in GLM have guaranteed convergence to global maxima? The top answer said that one can prove $\frac{d A}{d \theta}=\mathbb{E}[\phi(x)]$ $\frac{d^{2} A}{d \theta^{...
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1answer
36 views

While simulating the value of a double integral , why do we need to draw different samples everytime?

Suppose I want to simulate the value of the integral $\int_{0}^{1} \int_{2}^{3} 2xy \ dx dy$ using Monte Carlo methods. So, now, I draw a random sample from $U_1,U_2,...,U_n$ from $U(0,1)$ and for ...
-3
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1answer
62 views

Positive or negatively bounded CDFs [closed]

If $X\in\mathbb{R}^n$ is a continuous random variable whose cumulative distribution function is ordinarily $$F_X(x) = \int_{-\infty}^{\infty} f_X(x) dx $$ what is the meaning of $$F_X(x) = \int_{0}^{\...
2
votes
1answer
91 views

What is the fisher information matrix of the multivariate t distribution?

$\newcommand{\bx}{\mathbf{x}}$ $\newcommand{\bSigma}{\boldsymbol{\Sigma}}$ $\newcommand{\bE}{\mathbf{E}}$ $\newcommand{\bD}{\mathbf{D}}$ Consider the multivariate central t distribution with p.d.f. \...

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