All Questions
Tagged with integration pr.probability
133 questions
0
votes
1
answer
143
views
Formally confirm a formula for a certain three-dimensional constrained integral over the unit cube
The result of the three-dimensional constrained integration (for the Hilbert-Schmidt two-qubit absolute separability probability) over the unit cube $[0,1]^3$
\begin{equation} \label{one}
\int_0^1 \...
- 723
5
votes
1
answer
319
views
Spherical average of $\frac{1}{x}$
Let $X_1,...,X_n$ be points on $\mathbb S^1.$
We then define the expectation value $E(X)=\frac{1}{n}\sum_{i=1}^n X_i.$
Let $\frac{dS(X_1)}{2\pi}$ be the normalized surface measure of $\mathbb S^1,$ i....
- 124
0
votes
0
answers
146
views
Does the following sequence $\{g_n\}$ converge?
Consider a function sequence $\{f_n(t)\}$ ($n\in\mathbb{N}^+$) defined on the interval $(\frac{1}{2},1)$, where
\begin{eqnarray}\label{eqn:constraint1}
f_n(t)=\frac{\exp\left(n\left(\log R(h_t) - th_t\...
- 550
1
vote
2
answers
139
views
Inaccurate results for the analytical expression of $\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]$
I'm trying to plot a graph for the following expectation
$$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]=a 2^{-\frac{\kappa }{2}-1} b^{-\frac{\kappa }{2}} \theta ^{-\kappa } \...
0
votes
2
answers
246
views
Finding the expectation of $a \mathcal{Q} \left( \sqrt{b } \gamma \right) $, where $\gamma$ is a Gamma r.v
I'm trying to analytically find the following expectation
$$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right],$$
where $a$ and $b$ are constant values, $\mathcal{Q}$ is the ...
0
votes
1
answer
55
views
Looking for a family of random variables such that only the second clause is fulfilled [closed]
Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if
i) $sup_{i \in I} E(X_i) <\infty$
ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t....
- 11
12
votes
1
answer
628
views
A function with unexpectedly simple Legendre transformation
Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and
\begin{equation}
I(x)=
\begin{cases}
\frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\
\...
- 1,608
6
votes
1
answer
684
views
Differentiable dependence on the initial condition of the solution of a SDE
Let
$b,\sigma:\mathbb R\to\mathbb R$ be differentiable and Lipschitz continuous
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a complete and right-...
- 167
2
votes
1
answer
70
views
$ \int_{E}^{*}{\psi (t) d\mu(t)}=\int_{E}{\phi (t) d\mu(t)} $
Let $(T, \mathcal{A}, \mu)$ be an arbitrary measure space.
The outer integral over $(T, \mathcal{A}, \mu)$ of a (possibly nonmeasurable) function $\psi: T\to (-\infty, +\infty]$ is defined by:
$$
\...
- 199
0
votes
1
answer
86
views
Integral rising from difference of chi-squared random variables
Let $X,Y$ be independent random variables such that $X\sim\chi_{n-1}^{2}, Y\sim\chi_{1}^{2}$ are chi-squared distributed (where $n\geq2$ is a natural number). I am trying to evaluate $\mathbb{P}[X\leq ...
- 109
0
votes
0
answers
116
views
Finding a square integrable dominating function for function class
problem statement
For any $x \in R^d$, consider the function $$\phi(x,a) = \min_{1\leq j \leq k} \|x-a_j\|^2,$$
where $a = (a_1, \dots, a_k) \in R^{kd}$ and $\| \cdot \|$ is the $L_p$ norm for any $p ...
- 11
0
votes
1
answer
153
views
$ \{X_n\mathbb{1}_{X_n\in[-n,n]}\}$ is uniformly integrable
Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space.
Suppose $\{X_n\}$ is a sequence of random variables satisfying :
$$
\sup_{n}{\mathbb{E}(|X_n|)} <\infty
$$
Suppose that
$$
\dfrac{M_j}{...
- 57
1
vote
0
answers
222
views
L2 norm of the diagonal entries of a random rotation of a fixed matrix?
Let $X\in\mathbb{R}^{d\times d}$ be the diagonal matrix with $d/2$ entries equal to $1$ and $d/2$ entries equal to $-1$. Let $F_U \triangleq \frac{1}{d}\|\operatorname{diag}(U^{\dagger}XU)\|^2_F$ ...
- 91
0
votes
1
answer
102
views
Sign of expectation value
Consider a multivariate Gaussian-type measure $$d\lambda(x):=\nu_{\mu,\Sigma} e^{-\langle (x-\mu), \Sigma^{-1}(x-\mu) \rangle - \vert x \vert^2} $$
with vector $\mu \in \mathbb R^n$ and $\Sigma$ ...
- 536
0
votes
1
answer
135
views
Bounds on expectation of $X/(X^2 + c)$ with $X$ ~ Gaussian and $c > 0$
I'm trying to compute expectation of $X / (X^2 + c)$ when $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, and $c$ is some positive constant. I think this cannot be solved ...
- 11
3
votes
2
answers
168
views
A definite integral related to sample variances of bivariate Gaussians
This integral is needed to obtain the joint
distribution of the sample variances of a random sample from a bivariate
Gaussian distribution. For details on the joint distribution of the sample
means, ...
- 984
4
votes
2
answers
274
views
Is it true that the quantile function of an $L^1$ random variable is $L^2(]0,1[)$?
Let $(\Omega, \mathcal A, P)$ be a probability space. Let $X:\Omega \rightarrow \mathbb R$ be an $L^1(\Omega, \mathcal A, P)$ random variable.
We define the distribution function of $X$ by
$$F(x) = ...
- 143
0
votes
0
answers
115
views
Bayesian Bandits - What's the probability that choice K is the best?
I have $K$ very unfair coins. I don't know how unfair they are, but they all seem to have different probabilities of landing heads. I'd like to figure out which one is best as quickly as possible.
...
- 9
2
votes
1
answer
81
views
Perform certain constrained integrations over an ordered subsection of a 3-simplex, yielding "absolute separability" probabilities
Let us order the four points $\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq 0$ of a 3-simplex, $\lambda_1+\lambda_2+\lambda_3+\lambda_4=1$, giving us a subsection $L$.
Integration over $...
- 723
2
votes
1
answer
636
views
Sufficient condition for function of conditional probability density to be increasing
Let $Y$ and $W$ be two jointly distributed random variables; $Y$ takes values on $(y_1,y_2)$ and $W$ takes values on $(w_1,w_2)$. The conditional probability density of $W$ given $Y$ is given by $f_{W|...
- 143
6
votes
1
answer
433
views
Triangle inequality for Ito integral?
For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say
$$\left\vert\int_0^t f(s) \ ds\right\vert \le \Vert f \Vert_{\infty} \int_0^t \ ds$$
Now if ...
- 536
1
vote
2
answers
889
views
Simplify Wasserstein distance between Gaussians with binary cost function
Let $\mu_1$ and $\mu_2$ be 1D gaussian distributions with means $m_1$ and $m_2$ respectively and common variance $\sigma$. Let $\Omega$ be a closed subset of $\mathbb R^2$, and consider the cost ...
- 6,853
5
votes
3
answers
475
views
Closed form $\int_{0}^{\frac{r}{2}} {\binom{n}{p} \binom{n-p}{r-2p} 2^{r-2p}}{\binom{2n}{r}^{-1}} \ \text{d}p$
Note: This is exact copy of my Math.SE question, which I am reposting here, as despite bounty it did not receive any answers.
Let there be $n$ pairs of shoes in a box.
The the probability that from ...
- 67
5
votes
1
answer
2k
views
Normal multivariate orthant probabilities
(Previously I posted a similar question on math.SE, hoping that this question would have an easy answer. As the question appears hard, I am hoping I can perhaps get more feedback here.)
Let $\mathbf{...
- 733
2
votes
1
answer
803
views
On Riemann integration of stochastic processes of order $p$
Let $x:[a,b]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process, where $\Omega$ is the sample space from an underlying probability space. Let $L^p$ be the Lebesgue space of random variables on $...
user85801
1
vote
2
answers
818
views
Integral formula involving Legendre polynomial
I would like a proof of the following equation below, where $(P_n)$ denotes Legendre polynomials. I guess this formula to hold; I checked the first several values.
\begin{equation}
\int_{-1}^{1}\sqrt{...
- 31
2
votes
1
answer
544
views
Multiple Wiener-Ito integral distribution
Distribution of standard Ito integral is well known: $$I_1(f) = \int_0^T f(t)dB(t) \sim \mathcal{N}\bigg(0, \int_0^T f^2(t)dt\bigg).$$
Is it possible to find the distribution of multiple Wiener-Ito ...
8
votes
2
answers
330
views
q-Means and the mode of a distribution
Let $f:\mathbb{R} \rightarrow [0,\infty)$ be a continuous probability density function on $\mathbb{R}$ such that
\begin{equation}
\int_{\mathbb{R}} |x| f(x)\, dx < \infty,
\end{equation}
and ...
- 640
1
vote
1
answer
399
views
I want to disprove an equality involving a double integral
I want to show that the following equality does not hold:
\begin{equation}\label{at3}
\frac{\lambda^2-1}{2}x^2-\int_{-\infty}^{\infty}\!\!\int_{-\infty}^{\infty}K(y_1,y_2,x)\ln g(y_1,y_2)dy_2dy_1=...
- 19
2
votes
1
answer
2k
views
Explicitly representing a random variable in terms of indicator functions
Motivation:
I want to compute $$E[g(X)] := \int_{\Omega} g(X(\omega)) d\mathbb{P}(\omega) \tag{*}$$ without needing change of variable formula.
I want to prove the change of variable formula (you ...
- 247
14
votes
1
answer
2k
views
Why do we mainly integrate with respect to martingales?
Although my resarch focuses on PDEs (optimal transport, these days), I am currently trying to learn stochastic calculus and integration. I am just beginning in this topics, but I was wondering: why do ...
- 5,380
4
votes
1
answer
697
views
Numerical evaluation/approximation of non-central high-order moments of high-dimensional Gaussian measures?
I need to numerically evaluate/approximate non-central high-order moments of high-dimensional Gaussian measures/distributions with given mathematical expectations and covariance matrices.
The Gaussian ...
- 1,026
1
vote
0
answers
101
views
Density of Geometric Stable distribution
If we define
$$
\psi(t|\alpha, \beta, \gamma, \mu) = -it\mu+|\gamma t|^\alpha(1-i\beta \mathrm{sgn}(t) \Phi)
$$
with
$$\Phi = \tan \frac{\pi \alpha}{2} \mathbf{1}_{\{ \alpha \neq 1 \} } - \frac{2}{\pi}...
3
votes
1
answer
311
views
An integral by rough path.
If $(b, \mathbb{b})\in \mathcal{D}^{\alpha}[0,T],\ \alpha\in (\frac{1}{3}, \frac{1}{2})$. $\mathcal{D}^{\alpha}[0,T]$ is the space of those rough paths $(b,\mathbb{b})$
such that
$$ \|b\|_\alpha=...
- 515
2
votes
2
answers
1k
views
Closed-form solution for an integral involving the p.d.f. and c.d.f. of a $N(0,1)$-distributed random variable
Let $\phi(\cdot)$ and $\Phi(\cdot)$ be the probability and cumulative density functions, respectively, of a random variable with distribution $\text{N}(0,\,1)$. That is,
$$\forall x\in\mathbb{R}:\,\...
- 173
2
votes
1
answer
438
views
A dilemma about the definition of the stochastic integral $\int_a^b\Phi\:{\rm d}W$
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$T>0$
$(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration of $\mathcal A$
$W$ be an $\mathcal F$-Brownian motion on $(\Omega,\mathcal A,\...
- 167
2
votes
0
answers
207
views
Gaussian integrals and Showing $ \int f({\vec {x}})e^{\left(-{\frac {1}{2}}\sum \limits _{i,j=1}^{n}A_{ij}x_{i}x_{j}\right)}d^{n}x=e^{D}f|_{x=0}$
This is related to my other question on tackling a gaussian integral for $f(w,u)=\frac{1}{w-u}$.
Q1 Suggestions on evaluating gaussian integrals with "nice" functions (not necessarily polynomials)
...
- 5,474
3
votes
3
answers
593
views
An integral involving hyperbolic functions
I am wondering if it is possible to obtain a closed-form formula for
$$
f(\alpha) = \frac{1}{{\sqrt{2 \pi } \; \alpha }} \int^\infty_{-\infty} x^2 \cosh(x) \; e^{-\frac{\sinh ^2(x)}{2 \alpha ^2}} \...
-6
votes
2
answers
2k
views
Is there a transformation or a proof for these integrals?
Here are certain weighted Gaussian integrals I have encountered for which numerical computation reassures equality.
Question. Is this true? If so, is there an underlying transformation or just a ...
- 43.2k
-2
votes
1
answer
457
views
Expectation of random integral of deterministic function
Suppose I have some random variable $W$ along with its expectation $\mathbb{E}[W]$. My goal it to compute the integral
\begin{equation}
\mathbb{E}\left[\int_{0}^{W}f(t)dt\right] = \int_{0}^{\mathbb{E}...
- 48
14
votes
2
answers
1k
views
Expected number of lines meeting four given lines or "what is 1.72..."
Given four random lines in $\mathbb{R}P^3$, how many lines intersect all of those lines?
In the recent paper Probabilistic Schubert Calculus, Peter Bürgisser and Antonio Lerario
discuss this question ...
- 10.7k
106
votes
5
answers
10k
views
integral of a "sin-omial" coefficients=binomial
I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof?
For any pair of integers $n\geq k\geq0$, we have
$$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...
- 43.2k
3
votes
0
answers
228
views
Sub-multiplicative function in expectation or pointwise? [closed]
Consider the function that satisfies
$$ \mathbb{E}[f(X)f(Y)]\leq \mathbb{E}[f(XY)],$$
where $X\in\mathbb{R}$ and $Y\in\mathbb{R}$ are Gaussian random variables with mean $0$ and variance $1$, and ...
1
vote
0
answers
106
views
Improper integral of products and ratios of probability density functions
I am trying to find out whether the following integral is finite. The integrand consists of product of probability density functions.
$\int \frac{f(x_1,x_2, x_4^*)}{f(x_1^*,x_2, x^*_4)}\frac{f(x_1,...
- 11
0
votes
1
answer
503
views
Asymptotics of a 1D integral, or the orthant probability of an equicorrelated random Gaussian vector
Problem: Let $\phi(x)$ be the normal probability density function (pdf), and $\Phi(x)$ the normal cumulative distribution (cdf). I'm interested in the asymptotic behavior of the following integral
$I(...
- 968
2
votes
1
answer
207
views
Expectation of Truncated Bivariate Gaussian Random Variables
Suppose $Z , \epsilon \sim N(0, 1)$ are independent Gaussian random variables. Let $a \ll 1$ be a small positive number. Let $W = aZ + \epsilon$. It can be show that
\begin{align}
\mathbb{E} [ W^2 (Z^...
- 1,127
3
votes
1
answer
942
views
Cumulative integral of the Marchenko-Pastur density for Wishart eigenvalues
I can't find any work on the cumulative density function of the well-known Marchenko-Pastur density for the eigenvalues of a standard Wishart Matrix as its dimension goes to $\infty$, i.e.
$$F(\beta)=...
- 51
1
vote
1
answer
147
views
Proving that an integral related to order statistics is increasing in a certain parameter
Let $f$ and $F$ denote, respectively, the pdf and cdf of a probability distribution on $\mathbb R$. Take any natural $n\ge3$ and any real $a$ and $c$ such that $a\le c$.
Does it always follow that
$$...
- 13
2
votes
0
answers
254
views
Prove this function is increasing
I'm stuck in showing that the following function is increasing over the domain $\left[0,\hat{b}\right]$:
\begin{eqnarray}
\Pi\left(z\right) & = & \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}...
- 21
2
votes
0
answers
79
views
Compute Mixed Volume with Respect to Some Regular Sets
Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...
- 1,127