# 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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### 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

$\newcommand{\icol}{% inline column vector \left(\begin{smallmatrix}#1\end{smallmatrix}\right)% }$ Scenario: Let's consider a road segment on which there is continuous flow of cars circulating at ...
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### 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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### 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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### 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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### 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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### 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 ...