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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How do I show $$ \int_{0}^{1} exp(-f(x))dx=\sum_{n=0}^{\infty} \frac{(-1)^n}{n!} \int_{0}^{\infty} f(x)^n dx $$ [migrated]

I have been given a function f(x) defined on the interval $$[0;\infty[ $$ by the formula $$ \begin{array}{cc} \ f(x)= \{ & \begin{array}{cc}\ 0 & x= 0 \\ \ x\space log(x) ...
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Finding C for which f(x) is a density function

One of the points of the exercise states: Find the constant $C$ for which the following function is a density function $$ f(x)= \begin{cases} C(x-x^2) & 0 \leq x \leq 2\\ 0 ...
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How to solve for $k$ in integral? (hypothesis testing)

In a hypothesis testing problem I am stuck at this step, trying to solve for $k$ in the relation: $$F_{NegBin(n,0.5)}(k)=0.05$$ The problem is no matter which cumulative distribution function I ...
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Integrating indicator function to derive pseudovalue

I'm reading a paper on the use of pseudovalues in survival analysis and am trying to derive the pseudovalue for the restricted mean lifetime function. We have, $$ \hat{\mu}_{\tau_i} = \int_{0}^{\tau} ...
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Whether or not to integrate by parts an integral containing a derivative before solving it numerically [migrated]

I am estimating the derivative of an integral numerically. I believe that solving the integral by parts first before taking the derivative gives a more stable numerical result. However, I have some ...
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44 views

Solve $\int_{K}^{\infty}yf(y)\,dy$ where $y=A\exp\left(\sigma x-\frac{\sigma^2}{2}\right)$ and $x\sim N(0,1)$ [closed]

I need to solve $$\int_{K}^{\infty} y f(y)\,dy $$ where $y = A\exp\left(\sigma x-\frac{\sigma^2}{2}\right)$, $x\sim N(0,1)$, $K\geq0$. Any help?
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Gaussian closest to a maximum of Gaussians

Let $X_i \sim \mathcal{N}(\mu_i, \sigma_i)$ be independent, normally-distributed random variables. Let $$Y = a + b \max_i X_i$$ where $a \in \mathbb{R}$ and $b \in (0, 1)$. Which Gaussian ...
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18 views

Examples for integration estimator

suppose I'm interested in estimating $C=\int_{a}^{b}g(x)dx$, where $a$ and $b$ are known, and $g(x)=E(Y|X=x)$ is an unknown function of $x$. The data I have is $\{Y_{i},X_{i}\}_{i=1}^{n}$, then a ...
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Finding the MGF of a bivariate Normal Distribution [duplicate]

Given ($X$, $Y$) whose MGF is defined as: $$M_{XY}(s, t)=E[e^{sX+tY}]$$ Find $M_{XY}(s, t)$ when $X$ and $Y$ are two jointly normal random variables with $E[X]=\mu_X$, $E[Y]=\mu_Y$, $var(X)=\sigma^...
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Making a discrete probability question continuous

I'm trying to figure out how many coin flips you'd need to have a greater than 50% chance of having seen a heads, given a biased coin with heads probability $p$. From this question we can see that ...
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Integrate out the binary indicator variable in a two-sample ANOVA

I have two sets of data, A and B, that have unequal sizes, and I want to compare their means. The standard approach would be to do a t-test. Getting a little more sophisticated, we can think of that t-...
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44 views

The effect on expectation of biasing a distribution by a monotonic function

Let $$ x \sim g(x)$$ where $g(x)$ denotes the pdf of $x$. Let the pdf of another variable $x^*$ be denoted by $f(x^*)$ and let $$f(x^*) \propto g(x^*) z(x^*) $$ where $z(x^*)$ is a monotonic ...
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Analyse data assuming continuous process or discrete process

Assuming I have a dataset of temperature data sampled every 5 minutes and I want to find out its mean. If we assume that the data was sampled from a discrete process we can use the arithmetic mean: $\...
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Proof of identifiability

I have a random variable $R$ that takes values in ${1,2,3,4}$, and the conditional distribution of $R$ given each $(x_{1},x_{2})$ is given by the following formulas ($P_{i}(x_{1},x_{2})$ is just the ...
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Expectation and variance notation for ratio of random variables [duplicate]

When we have a ratio of random variables, is their expectation/variance defined in the same way? That is, if we want to write out explicitly $E[\frac{X}{Y}]$ where X and Y are random variables, then ...
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23 views

DeGroot P.155 integration problem for multivariate distributions

I am stuck with the integral for equation 3.7.4 and do not see how it was done. Could someone provide me with some hints or resources to read around?
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46 views

Integration over time

Can someone explain this step : $\frac{1}{t_n-t_1}\sum_{i=1}^{n-1} \int_{t_i}^{t_{i+1}}[\theta_i + \frac{\theta_{i+1}-\theta_{i}}{t_{i+1}-t_i}(t-t_i)]dt $ to $\frac{1}{t_n-t_1}\sum_{i=1}^{n-1}\frac{...
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Nested Uniform Distributions in Monte Carlo Integration

In terms of importance sampling for numerical Monte Carlo integration we can proceed as follows: \begin{align} \int_{\Omega} p(\mathbf{x}) d\mathbf{x} &= \int_{\Omega} p(\mathbf{x}) \frac{q(\...
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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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1answer
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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\...
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178 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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57 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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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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62 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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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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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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163 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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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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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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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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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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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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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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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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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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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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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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How to compute $E[ (|X|) X]$ when $X \sim N(0,1)$?

Any help would be appreciated.
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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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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
108 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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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
371 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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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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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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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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32 views

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

How to integrate the following? Thanks
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30 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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3answers
110 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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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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