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/

Filter by
Sorted by
Tagged with
0
votes
0answers
12 views

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 ...
0
votes
1answer
12 views

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 ...
0
votes
1answer
23 views

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 ...
2
votes
0answers
61 views

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 ...
0
votes
1answer
38 views

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 ...
0
votes
1answer
44 views

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_{-...
0
votes
0answers
15 views

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
0
votes
0answers
31 views

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(\...
8
votes
3answers
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 ...
0
votes
1answer
45 views

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, $X$ ...
4
votes
1answer
57 views

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) = \...
4
votes
1answer
204 views

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(\...
0
votes
1answer
56 views

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\}...
3
votes
1answer
44 views

$\int_{-\infty}^{\infty} x^3 f(x) dx < \infty$ 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)=...
2
votes
0answers
87 views

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 ...
0
votes
0answers
11 views

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}^{...
1
vote
0answers
58 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 ...
4
votes
0answers
91 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....
1
vote
1answer
70 views

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 ...
9
votes
2answers
860 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 $\...
0
votes
0answers
31 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-\...
5
votes
1answer
60 views

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 $$\int_x (x - E(X)) E(Y|X) f_xdx = \int_x x E(Y|X) f_x dx - E(X) \int_x E(Y|X) f_x dx .$$...
1
vote
1answer
34 views

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{\...
1
vote
2answers
51 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 $x$...
2
votes
1answer
204 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[...
1
vote
1answer
23 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 ...
2
votes
1answer
85 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 ...
1
vote
1answer
120 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 ...
0
votes
0answers
41 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 ...
0
votes
0answers
50 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 ...
0
votes
0answers
79 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
votes
1answer
37 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 ...
0
votes
0answers
129 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 ...
0
votes
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 ...
3
votes
1answer
64 views

$f$ is a decreasing function whose integral converges. Does $\lim_{x \to \infty}xf(x) = 0$?

My finals are over and I cannot help but ruminate over this particular problem. Could anyone help prove this? Suppose $f$ is a continuous decreasing function on $[0,\infty)$ and $\int_0^\infty f(t)\, ...
2
votes
2answers
731 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. ...
4
votes
0answers
167 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
votes
1answer
130 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 (...
0
votes
0answers
19 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 ...
1
vote
0answers
91 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 ...
0
votes
0answers
9 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 ...
0
votes
0answers
12 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 ...
0
votes
0answers
23 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_{...
1
vote
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\...
10
votes
1answer
286 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 ...
1
vote
2answers
88 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)^...
0
votes
0answers
27 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
votes
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. $$ \int_{y=-\infty}^{y=\infty} y f(x,y) dy$$
8
votes
1answer
483 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:...
-1
votes
1answer
102 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{...

1
2 3 4 5
7