# 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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### 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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### 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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### 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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I recently read a biostats paper which featured the following identity: $$\sum_{y, l, m} y P(y, l, m \mid c, a) \frac{P(l \mid a, c) P\left(m \mid a^{*}, c\right)}{P(l, m \mid c, a)}=E\left(Y \frac{P\... 1answer 286 views ### Plain English explanation of Ito's integral? I'm looking for a plain English explanation of Ito's integral. I don't need an exhaustive proof, derivation, etc. Just a simple ~this is effectively what it does and why it's better than a Riemann sum ... 2answers 88 views ### Marginal distributions given the distribution of range I'm working with an upper diagonal distribution whose distance from the diagonal is Lomax Pareto (Type II) distribution. The distance of a point from the diagonal line y = x is \frac{\sqrt{(x_0-y_0)^... 0answers 27 views ### How can integral be interpreted as linear model? I've searched how can I apply linear transformation to following problem, but it seems like there's just not intuitive explanation enough for me. We have$$ p\left(\mathbf{v}\mid \mathbf{h}\right) = \...
Does the integral below have a specific interpretation in statistics? It looks like the marginal expectation of y, without integrating over x. $$\int_{y=-\infty}^{y=\infty} y f(x,y) dy$$