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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Solving integral with generic pdf [closed]

I have a continuously differentiable function $h:[\underline{x},\overline{x}] \rightarrow \mathbb{R}$, and a continuous random variable $X$ distributed according to a cdf $F$, with full support on $[\...
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Bayesian statistics: what is the variable we are integrating in?

This is a screenshot from Bayesian Data Analysis by Gelman. I am a little bit confused by Equation 1.4 (first and second lines), having read Equation 1.3. In Equation 1.3, the variable of integration ...
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Help for understanding Leibniz Integral Rule [migrated]

Why is it that according to Leibniz Integral Rule, we cannot take derivative inside integration when both derivative and integration are over the same variable? For example: Let's say we have $f(x,t)$....
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Single-sample self-normalized importance weighting

Self-normalizing sampling schemes (https://artowen.su.domains/mc/Ch-var-is.pdf) seem to require at least two samples to give non-trivial weightings under an importance sampling distribution. Is there ...
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transformation of a kernel density estimate to uniform distribution

I am interested in estimating the expected value of a function, $f(x)$ with respect to a probability density function, $P(x)$. I am exploring a method that requires I change variables from the ...
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For $Y \geq 0$, prove that $Pr(Y \geq k) \leq E(Y)/k$

Let $Y$ be a non-negative random variable, $k$ be any positive constant, show that $Pr(Y \geq k) \leq E(Y)/k$. My attempt (using integration by parts): \begin{align} \int_0^k y \,dF(y) &\leq E(Y) \...
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Multivariate Gaussian probability mass inside a sphere

Assume I have some d-dimensional multivariate gaussian $X\sim\mathcal{N}\left(\mu,\Sigma\right)$ and some sphere $C=\left\{ x:\left\Vert x-z\right\Vert_2\le r\right\}\subseteq\mathbb{R}^{d}$. I was ...
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Finding E(XY) for joint probability density

$Joint \:probability\;f(x,y) = 2/3 \:for\: 0 < x < 1, 0 < y < 2, x < y, and\: 0\: otherwise $ $E(XY)=\int_{0}^{1}\int_{x}^{2} \frac{2}{3}xy \:dy \:dx = \frac{7}{12} - (1)$ $E(XY)=\...
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Solve an inequality finding the upper bound

Suppose that there exists a constant $C$ such that the following relation holds for all $G$: \begin{equation*} \vert T(F)-T(G) \vert \le C \sup_y \vert F(y)-G(y) \vert \end{equation*} Suppose that ...
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Expected value of the largest order statistic for $Uniform(\theta,2\theta)$ [duplicate]

I'm struggling to find when $X_1,\ldots,X_n \sim Uniform(\theta,2\theta)$, how the expected value of the largest order statistic is $E[X_{(n)}]=\dfrac{2n+1}{n+1}\theta$. I can find that the density of ...
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Probability of collision: mathematical vs probabilistic modeling

$\newcommand{\icol}[1]{% inline column vector \left(\begin{smallmatrix}#1\end{smallmatrix}\right)% }$ Scenario: Let's consider a road segment on which there is continuous flow of cars circulating at ...
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Location-scale parameter with non-informative (improper) prior : at what condition is the posterior proper?

Consider the setup: Let $(X_i | \mu = m, \sigma = s)$ be a continuous random variable with pdf$$f_{X_i | \mu, \sigma}(x | m, s) = f_{X_i | \mu , \sigma}\big( \frac{x-m}{s} | 0,1 \big) \ s^{-1}, x \in \...
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How to show that this integral of the normal distribution is finite?

Numerically, I have noticed that $$\int_{-\infty}^{\infty} \dfrac{\phi(x)^2}{\Phi(x)}dx < \infty$$ where $\phi$ and $\Phi$ are the standard normal pdf and cdf. However, I do not see how to prove it....
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Integrating out random effect from hierarchical model

Given $Y_{ij}$, $M_j$, and $R_i$ as observables, my probablistic model is $$ Y_{ij} = \theta_{i}M_j + \varepsilon_{ij}\\ \theta_{i} = \beta R_i + \phi_{i} \\ \varepsilon_{ij} \sim N(0, \sigma), \...
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integration of product of a gaussian pdf and a student-t pdf

I want to perform the following integration wrt $x$: $$\int_{-\infty}^{\infty}\frac{1}{\sqrt(2\pi\sigma^2)}e^(\frac{-(y-hx)^2}{2\sigma^2})[(1+\frac{x^2}{b})^{-(\frac{b+1}{2})}]dx$$ Here first part is ...
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HOW to determine $S$ in $SEIR$ epidemiology model

With the contact rate parameters for a $SEIR$ model as follows: $$ \begin{aligned} \beta &= 0.139\\ \gamma &= \frac{1}{10.4}\\ \sigma &= 0.1\\ R_0 &= 1.4456\\ N &= 3,787,000\\ I(0) ...
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Positive definiteness of integral of matrix

I was reading a paper, and did not understand a statement that the author made without further explanation. The author derives the limiting distribution of a non-linear least-squares estimator and ...
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How is Nadayara Watson KDE proof

I was looking at Wikipedia article on Nadayara-Watson Kernel regression section, in the proof part they state But I'm having trouble understand why: Turns to just yi. Sorry I'm missing something so ...
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Law of total probability for random variables with Y < X

Could someone explain to me why the following equation holds? It is related to the law of total probability , but I don't get it. I'm confused because it's two random variables on the right side and ...
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2 votes
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For a general multivariate normally distributed $\boldsymbol{X}$, what is the expectation of $1/(\boldsymbol{X}^T \boldsymbol{X})$

For $\boldsymbol{X} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, where $\boldsymbol{\mu} \in \mathbb{R}^N$, $\boldsymbol{\Sigma} \in \mathbb{R}^{N \times N}$ is positive definite, how to ...
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Indicator function with equal sign for probability integral

In the beginning of the book Train (2009, p.4) on "Discrete choice methods with simulation" we read: Define an indicator function $I[h(x,ε) = y]$ that takes the value of $1$ when the ...
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Moments of $\text{exp}(-|x|^{1/2})$

I'm supposed to show that all of the moments of the density $\text{exp}(-|x|^{1/2})$ are finite. I'm not convinced this is true though. The $p$th moment is \begin{align*} \mathbb{E}[X^p] &= \int_{-...
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Expansion of CDF of normal distribution using integration by parts

How does the author express last F(x) mathematical expression in terms of second last H(x) mathematical expression
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Expected value of 1/(1−X) of a Gamma distribution [duplicate]

I was interested in calculating $E\left(\dfrac{1}{1-X}\right)$ where $X\sim$ Gamma ($n,\lambda$), but I wasn't able to solve the associated integral using standard integration techniques. $$E\left(\...
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8 votes
3 answers
2k views

Monte Carlo simulations for arbitrary functions

I'm familiar with MC methods for approximating PDF integrals. But in this question, I'm curious how we might adapt these methods for other problems. For example evaluating $\int_{0}^{1} x^2 dx$ . I ...
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A proof related to an expected value (revised)

I uploaded a question asking how to proof an equation. But, I felt that I made some confusions, and I will ask the question in a more tidy form with details. Suppose that $X \sim N(0,c)$. That is, $X$ ...
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Finding the value of $k$ for an Uniform Distribution defined on $(-k,k)$

If $X$ be an uniform distribution defined on $(-k,k)$, then the value of $k$ for so that : $$P(|X|<1) = P(|X|>2)$$ I began by defining the $p.d.f$ of the Uniform function namely: $$ f(x) = \...
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CDF of Dirichlet Distribution

We know that a random variable $p=(p_{1}, p_{2},..., p_{K})$ which follows a $\textit{Dirichlet}$ distribution with parameters $\textbf{a} = (a_{1}, a_{2},..., a_{K})$ has as pdf $$f(p) = \frac{1}{B(\...
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0 votes
1 answer
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Integrating with considering two indicator function

Consider exponential random variables $X$, $Y$, and $Z$ with $\lambda_x$, $\lambda_y$, and $\lambda_z$, respectively. Now I want to calculate the following integration: $$E[X1_{\{X<Y\}}1_{\{X<Z\}...
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3 votes
1 answer
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$\int_{-\infty}^{\infty} x^3 f(x) dx < \infty$ then $Cov(X,X)<\infty$ ? TRUE OR FALSE

$x \in R$ is a continuous random variable. Is the statement : IF $\int_{-\infty}^{\infty} x^3 f(x) dx < \infty$ then: $Cov(X,X)<\infty$ .TRUE? My thought was that Var(x)=Cov(x,x) , so $Var(x)=...
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2 votes
0 answers
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How to infer a form of the function by observing its integral values?

Let me start with giving an analogy: suppose we have a plant, which grows (gains mass) depending on weather conditions (solar radiation, $x$) and its age/vegetation phase $a$. We can only observe its ...
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Integral of the Survival Function multiplied by the Distribution Function raised to a real power

Let $ F $ be a continuous life distribution with survival function $ \bar{F},$ density $ f $ and finite mean $ \mu. $ While doing some calculations, I came up with the following integral $$I=\int_{0}^{...
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1 vote
0 answers
61 views

Bayesian Parameter estimation (Pattern Classification by Duda, et al

I have been trying to solve question 17 of chapter 3 (Maximum Likelihood and bayesian estimation) of the book "Pattern Classification" by Duda, et al. The question goes as follows: Now the ...
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6 votes
0 answers
107 views

Lebesgue-Stieltjes integration by parts on a half-open interval

I have run into a problem in a proof of the bound for the rate convergence of an empirical risk function based on unbounded loss to the true model risk (Vapnik, Statistical Learning Theory, Theorem 5....
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1 answer
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Decompound a Compound Probability Distribution

I am trying to figure out how to deconvolve or decompound a compound probability density function - knowing one of the distributions and having samples from the compound distribution. Assume I only ...
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9 votes
2 answers
899 views

Is it wrong to say that a Riemann sum is an unbiased estimate of an integral?

Would it be wrong to say that a Riemann sum approximation of an integral \begin{align} \int_a^b f(t) \mathrm{d}t \approx \sum_{k=1}^{n_\text{samples}} f(t^{\ast}_k)\Delta t, \end{align} where $\...
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0 votes
0 answers
32 views

Integral of the error function

Assume a pair of normal variables $a,b\sim N(\mu, \Sigma)$, with $\rho_{ab}\neq0$. We know their joint distribution in the (shorthand) form: $$f_X(a,b)=\frac{1}{2\pi\sqrt{|\Sigma|}}exp(-\frac{1}{2}(x-\...
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5 votes
1 answer
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How can $E(X)$be factored out of integration?

Recently, I posted a question on this forum here. In the answer to the question, it was posted suggesting that $$\int_x (x - E(X)) E(Y|X) f_xdx = \int_x x E(Y|X) f_x dx - E(X) \int_x E(Y|X) f_x dx .$$...
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1 vote
1 answer
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Derivative of a Bivariate normal CDF with respect to its variables

Following up on the question (and answers) here, I'm trying to derive $\frac{\partial \Phi(x_1, x_2|\mathbf{\underline{\theta}})}{\partial x_1}$ and $\frac{\partial \Phi(x_1, x_2|\mathbf{\underline{\...
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1 vote
2 answers
167 views

Generating a Random Value Vector from an Exponential Distribution using R

Given a standard PDF of the form $f(x)=ae^{-ax}$ with domain $[0,+\infty)$, its CDF being $F(x)=1-e^{-ax}$, and a mutated CDF that takes $p \in [0,1]$ as a probability and returns the corresponding $x$...
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2 votes
1 answer
223 views

Weighted Average and Expectation in machine learning

Bishop's book defines expectation as "weighted average of a function". $$E[f(x)] = \int f(x)p(x)\text dx$$ However, the Wikipedia page of weighted function defines a weighted average as $$E[...
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1 vote
1 answer
26 views

Integrating in log-space with a change of variable

I have a probability density function $P(f|\mu,\sigma) = \mathcal{N}(f|\mu,\sigma)$. I need to change the variable $f$ to $L = \log_{10}[f]$ so I can integrate it jointly with another PDF whose domain ...
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2 votes
1 answer
161 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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1 vote
1 answer
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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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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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0 answers
71 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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112 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 ...
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2 votes
1 answer
41 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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0 votes
0 answers
160 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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0 votes
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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 ...
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