Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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 can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw$

Suppose $\phi(\cdot)$ and $\Phi(\cdot)$ are density function and distribution function of the standard normal distribution. How can one calculate the integral: $$\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw$$
hadisanji
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"The total area underneath a probability density function is 1" - relative to what?

Conceptually I grasp the meaning of the phrase "the total area underneath a PDF is 1". It should mean that the chances of the outcome being in the total interval of possibilities is 100%. But I cannot really understand it from a "geometric" point of…
TheChymera
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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. I would appreciate any hints.
finit
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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 is 0.5, while if you take the log of that sample, then take the mean of the…
Jeremy Dorner
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What is the expected value of modified Dirichlet distribution? (integration problem)

It is easy to produce a random variable with Dirichlet distribution using Gamma variables with the same scale parameter. If: $ X_i \sim \text{Gamma}(\alpha_i, \beta) $ Then: $ \left(\frac{X_1}{\sum_j X_j},\; \ldots\; , \frac{X_n}{\sum_j X_j}\right)…
Łukasz Lew
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Integrating an empirical CDF

I have an empirical distribution $G(x)$. I calculate it as follows x <- seq(0, 1000, 0.1) g <- ecdf(var1) G <- g(x) I denote $h(x) = dG/dx$, i.e., $h$ is the pdf while $G$ is the cdf. I now want to solve an equation for the upper limit…
user46768
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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 (or insert other numeric approximation.) With…
jbuddy_13
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Vector calculus in statistics

I'm teaching a class on integration of functions of several variables and vector calculus this semester. The class is made up most of economics majors and engineering majors, with a smattering of math and physics folks as well. I taught this class…
Paul Siegel
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Expected log value of noncentral exponential distribution

Suppose $X$ is non-central exponentially distributed with location $k$ and rate $\lambda$. Then, what is $E(\log(X))$. I know that for $k=0$, the answer is $-\log(\lambda) - \gamma$ where $\gamma$ is the Euler-Mascheroni constant. What about when…
Neil G
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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 $\Delta t = \left(b - a\right)/n_\text{samples}$,…
themainhatch
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Integral identity of lemma contained in infoGAN paper

I've come across a lemma in the infoGAN paper. I do not understand the derivation of Lemma 5.1 in the addendum of the paper. It goes as follows (included as png): I do not understand the last step. Why can one pull $f(x,y)$ into the inner-most…
spurra
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Example of computing the expectation of a discrete RV using Riemann-Stieltjes integral?

Riemann-Stieltjes integral notation is used in expectation expressions in some probability texts. Basically, dF(x) pops up in the integral rather than f(x)dx in the integral, since the CDF F(x) may not be differentiable for a discrete distribution.…
frelk
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Gibbs Sampler transition kernel

Let $\pi$ be the target distribution on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R^d}))$ which is absolutely continuously wrt to the $d$-dimensional Lebesgue measure, i.e : $\pi$ admits a density $\pi(x_1,...,x_d)$ wrt to $\lambda^d$…
user2016445
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Expectation of von Mises Fisher Distribution

The von Mises- Fisher distribution is defined as $$ \frac{\kappa^{p/2-1}}{2\pi I_{p/2-1}(\kappa)}\exp(\kappa \mu^Tx) $$ It is defined over the unit sphere i.e. $||x||_2^2=1$. My question is what is $E(x)$. I got a feeling it's simply $\mu$ but how…
sachinruk
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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 a constant speed $V_{car_1}$ and where cars are…
Benoit Fgt
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