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Proving bound on expectation of likelihood ratio involving mixtures

Let $p$ be a Lebesgue density function with infinite support (i.e. $p(x)>0 \forall x\in \mathbb{R}$ and $\int p(x) dx = 1$). Moreover, assume that $p$ is even (i.e. $p(x) = p(-x)$) and unimodal: $p(...
ILoveMath's user avatar
0 votes
0 answers
36 views

Contribution of Fisher information near jump points in convolved probability distributions

I am trying to compute the contribution to the Fisher information from jump points $b_i(\theta)$ in the convolved function $f(x; \theta)$ with respect to the parameter $\theta$. I am unsure whether it ...
Luna Belle's user avatar
1 vote
1 answer
62 views

MGF relevant to modified 2nd kind Bessel

Given the moment-generating function $$ m_{0}(t)=\frac{1}{\sqrt{1-t^2}}\,\text{ for }t<1, $$ which corresponds to a distribution with density $$ f(u) = \frac{1}{\pi}K_{0}(\frac{u}{\pi }) $$ where $...
Akiyoshi TOMIHARI's user avatar
2 votes
0 answers
104 views

Existence of Dirac measures in the context of joint and marginal distributions

Let $\pi$ be the joint law of $(X, Y)$ with marginal distributions $\mu$ and $\nu$. We assume that we have: for all $A \in \mathcal{B}(\mathbb{R})$ such that $\mu(A) > 0$ $$ \nu\left(\{y \in \...
thibault jeannin's user avatar
0 votes
0 answers
149 views

Reference book for a probability course

In the next months I am planning to deliver a (more-or-less) advanced course in probability theory. My students will have had already a first encounter with discrete probability theory (discrete ...
Johnny Cage's user avatar
  • 1,561
2 votes
0 answers
124 views

Generalization of the triangle inequality to complex exponents: What is $P\left(\left| x^{a+bi} + y^{a+bi} \right| \ge \left|z^{a+bi}\right|\right)$?

Let $x \le y \le z$ be the length of the sides of a triangle whose vertices are uniformly random on the circumference of a circle. In this question, it was proved that if $a \ge 1$, then the ...
Nilotpal Kanti Sinha's user avatar
13 votes
1 answer
762 views

If $(a,b,c)$ are the sides of a triangle, then the probability $P(ax + by \ge c) = \frac{4}{\pi^2}\chi_2(x) + \frac{4}{\pi^2}\chi_2(y)$

Posting this question in MO since it is unanswered in MSE Let $(a,b,c)$ be the side of a triangle. In its most general linear form, the triangle inequality can be expressed as: Does $ax + by \ge c$ ...
Nilotpal Kanti Sinha's user avatar
11 votes
1 answer
746 views

Find the area of the region enclosed by $\sin^p x+\sin^p y=\sin^p(x+y)$, the $x$-axis and the $y$-axis (comes from a probability question)

Consider the graph of $\sin^p x+\sin^p y=\sin^p(x+y)$, where $x$ and $y$ are acute, and $p>1$. Here are examples with, from left to right, $p=1.05,\space 1.25,\space 2,\space 4,\space 100$. Find ...
Dan's user avatar
  • 3,527
3 votes
0 answers
105 views

Maximal-type inequality for a Borel probability measure supported on a subset of $L^2(\mathbb{R}^d)$

Let $\mu$ be a Borel probability measure on $L^2(\mathbb{R}^d)$ for $d\ge 1$ which is moreover supported on the unit sphere $$S=\{\phi\in L^2(\mathbb{R}^d): \| \phi\|_{ L^2(\mathbb{R}^d)}=1\}.$$ Let ...
Dispersion's user avatar
25 votes
5 answers
2k views

Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\sin y}{\sin(x+y)}$ and the $x$-axis (comes from a probability question)

This question resisted attacks at MSE, so I am posting it here. Here is the graph of $\dfrac{\sin x}{\sin y}=\dfrac{\sin x+\sin y}{\sin(x+y)}$. Find the area of the region enclosed by the curve and ...
Dan's user avatar
  • 3,527
3 votes
1 answer
271 views

Expectation on a Polish space

I was wondering, if given a Polish space $X$, and given some probability measure $p$ on $X$, can the expectation of an $X$-valued function be taken? In particular, would the integral $\int_X x dp$ ...
J.R.'s user avatar
  • 291
2 votes
1 answer
122 views

Analytical solution for a double integral involving logistic functions and Gaussian distributions

I am working on a mathematical problem involving the evaluation of a double integral, and I am seeking an analytical solution or techniques to solve it. The integral I'm dealing with is as follows: ​$$...
Charles's user avatar
  • 31
1 vote
0 answers
87 views

$f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?

Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...
vaoy's user avatar
  • 309
4 votes
1 answer
205 views

Show that $\frac{1}{2 \pi i} \oint_{\mathbb{S}^1} \frac{1-\hat{f}(\xi)}{1-\xi}\cdot \frac{\mathrm{d} \xi}{\xi^{n+1}} \to 0$ as $n \to \infty$

Let $f = (f_0,f_1,\ldots,f_n,\ldots) \in \mathcal{P}(\mathbb N)$ be a probability distribution on $\mathbb N$ and denote by $$\hat{f}(z) = \sum_{n\geq 0} z^n f_n$$ for its probability generating ...
Fei Cao's user avatar
  • 730
3 votes
1 answer
205 views

Bound on an integral representing a difference of two relative entropies

Let $ f : [0,1] \to \mathbb{R} $ be a function satisfying: 1.) $ |f(x)| \leqslant a $ for some $ a < 1 $, and 2.) $ \int_0^1 f(x) {\mathrm d}x = 0 $. I would like to know whether the following ...
aleph's user avatar
  • 503
1 vote
2 answers
163 views

Integral with linear function, Normal PDF, Normal CDF

I am trying to calculate the following integral: $$\int_a^\infty x \Phi(cx+d) \phi\left(\frac{x-\mu}{\sigma}\right) dx,$$ where $\Phi$, $\phi$ denote the CDF and PDF of the standard Normal $N(0,1)$. I ...
Margot.'s user avatar
  • 49
4 votes
1 answer
424 views

An exercise on log-concave random variable on the real line

Let $X$ be a real random variable with log-concave density $f$. Assume that $E(X) =0$ and $E(X^2)=1$. Show that there is a universal (independent of $X$) constant $c>0$ such that: $$P(X\in[-1/2;0])\...
Gericault's user avatar
  • 245
1 vote
1 answer
152 views

The monotonicity of the bivariate normal with non-isotropic covariance

Let $Y = (Y_1, Y_2) \sim N(0, 11^T + I)$, be a bivariate normal random variable with non-isotropic covariance. Define $y = (y_1, y_2)$ and let \begin{align} F_{\delta}(y) = \Pr[Y_1 > y_1 - \delta, ...
Jon Lebensold's user avatar
-1 votes
1 answer
990 views

Random variable as an integral of an indicator function

This answer says that if $X$ is a random variable and $X_+ = \mathrm{max}(0, X)$, then $X_+ = \int_0^\infty I_{\{X > x\}}\mathrm{d}x$. I'd like to know how to derive this starting with $A \in \...
johnsmith's user avatar
  • 115
2 votes
1 answer
141 views

Injectivity of two sided Laplace transform

Let $\mu,\nu$ be finite Borel measures on $\mathbb R$. Assume that there is an open interval $(a,b)$ on which the Laplace transforms exist and coincide: $$ \int_{-\infty}^\infty e^{-tx}\,d\mu(x) = \...
Lau's user avatar
  • 769
0 votes
1 answer
222 views

Condition for $f^\prime$ to be absolute integrable

Suppose $f(x)$ is the probability density function of a random variable $X$, which means: $$\int_{a}^{b} f(x) dx = 1$$ Also suppose $f$ is continuous and differentiable. Provide a non-trivial ...
Mingzhou Liu's user avatar
30 votes
4 answers
2k views

If $X$ and $Y$ independent and identically distributed, then $E(|X-Y|)\leq E(|X+Y|)$. Are other proofs of this known?

I know a proof of the theorem that if $X$ and $Y$ independent and identically distributed, then $E(|X-Y|)\leq E(|X+Y|)$. The proof uses an integral representation of the absolute value, $$\int_0^\...
janis's user avatar
  • 409
3 votes
1 answer
146 views

Convolution between normal distribution and the maximum over $m$ Gaussian draws

$\DeclareMathOperator\erf{erf}$ Let's consider the Gaussian distribution $P_X(x)= \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{x^2}{2 \sigma^2}}$. Now consider the random variable $W \equiv \max \{ X_1, ...
user1172131's user avatar
3 votes
0 answers
176 views

What is the meaning of big-O of a random variable?

I encountered this problem in a book "Pattern Recognition and Machine Learning" by Christopher M. Bishop. I excerpt it below: screenshot of the book In the excerpt, the big-O notation $O(\xi^...
zzzhhh's user avatar
  • 31
1 vote
1 answer
199 views

Probability of multivariant gaussian random variables in different areas

$\newcommand{\sgn}{\operatorname{sgn}}$Let $X_i$ is a gaussian random variable correlated with others. we want to find the probability of each possible case to find the expectation of following ...
A. R.'s user avatar
  • 25
4 votes
1 answer
136 views

Decreasing tail integrals for nonnegative random variable $X$

Let $X$ be a nonnegative random variable with density function $f(x)$, distribution function $F(x)$, survival function $S(x)=1-F(x)$ and finite first and second moments. Let also $$\ell(x):=\frac{1}{...
Jimmy R.'s user avatar
0 votes
1 answer
205 views

Inhomogeneous Markov chains and the product-integral as a solution to the Kolmogorov forward equation

We have a inhomogeneous continous $K$-State Markov chain $X(t)$ with transition intensity matrix $Q(t)$. Therefore its entries are: $$q_{ij}(t)= \lim_{\delta \to 0} \frac{1}{\delta} \mathbb{P}(X(t+\...
Bloble's user avatar
  • 3
1 vote
1 answer
241 views

Integration by parts for indicator of a sphere to indicator of a ball

Broadly speaking, I have a radial distribution on $\mathbb R^n$, i.e., the pdf only depends on the $\ell_2$-norm of the argument. I would like to obtain an expression for the pdf in the form $\int_{w=...
Nicolas Resch's user avatar
2 votes
0 answers
136 views

Multiple integral with diagonal constraint (short-range)

I am looking for an upper bound on the following integral: $$\int_{X_{\delta}}\prod_{j\neq i=1}^{n}\left ( \frac{\delta}{\min (\max(\epsilon, |a_i-a_j|),\delta)}\right )^{b} \prod_{i=1}^{n} da_{i},$$ ...
Thomas Kojar's user avatar
  • 5,474
2 votes
1 answer
330 views

Probability density of a hyperplane for a Gaussian distribution

I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
etal's user avatar
  • 162
2 votes
1 answer
173 views

Radon transform of the function $h(x_1,\ldots,x_n) = x_1 g(x_1,\ldots,x_n)$, where $g$ is the density of multivariate Gaussian $N(\mu,\Sigma)$

Given an absolutely integrable function $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined for every $(w,b) \in (\mathbb R^n \setminus \{0\}) \times \mathbb R$ by $$ R[f](w,b) := ...
dohmatob's user avatar
  • 6,853
0 votes
1 answer
370 views

Closed-form CDF for bivariate normal distribution in point $(\Phi^{-1}(p),\,\Phi^{-1}(p))$

Let $\Phi(x)$ be a CDF of standard normal distribution and $\Phi^{-1}(p),\,p\in(0,1)$ its inverse. It is evident that $$ \mathbb{P}(X<\Phi^{-1}(p))=\Phi(\Phi^{-1}(p))=p, $$ where $X\sim N(0,1)$. Is ...
Fancier of Mathematica's user avatar
0 votes
1 answer
243 views

Integral form of expectation with respect to complex random variables [closed]

Let $h$ be a random variable and $g(h)$ be a real-valued function of $h$. We know that if h is a real-random variable then: $E_h[g(h)] = \int_{-\infty}^{\infty} f(h) g(h) dh$ where f(h) is the PDF of ...
Bertrille's user avatar
1 vote
1 answer
613 views

Integral of the product of a gaussian pdf and cdf

I am trying to solve the integral of a gaussian cumulative distribution function and a gaussian probability function. On this site I have seen solutions of similar, less general integrals (e.g. ...
Kurt Z.'s user avatar
  • 11
2 votes
1 answer
119 views

Exact formula or non-trivial upper bound on p-norm of $f(x)=\|x\|_2$ in $[0,1)^d$

I wonder whether one can exactly calculate the following integral in terms of $d$ and $p\geq 1$ or not, or a better bound(than the trivial one I am going to give) in terms of $d,p$: $$\left(\int_{[0,1)...
MikeG's user avatar
  • 715
4 votes
2 answers
316 views

Average value of $\frac{x'A^2x}{x'A^3x}$ over surface of $n$-dimensional sphere

Suppose $A$ is a diagonal matrix with eigenvalues $1,\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n}$ and $x$ is drawn from standard Gaussian in $n$ dimensions. Define $z_n$ as follows $$z_n=E_{x\sim \...
Yaroslav Bulatov's user avatar
4 votes
0 answers
75 views

Marginalization of Wishart distribution

Consider the following Wishart distribution $$ f({\bf W}) = \frac{ |{\bf W}|^{(n-p-1)/2} \exp\big[-\frac{1}{2}\text{tr}({\bf V}^{-1}{\bf W} ) \big] }{2^{np/2} |{\bf V}| \Gamma_p(\frac{n}{2})} \tag{1} $...
RenatoRenatoRenato's user avatar
2 votes
1 answer
102 views

Approximation of $\Phi (p)$

I am trying to find the asymptotic behavior (with respect to N) of the integral $$ \frac{2}{\sqrt{\pi}}\int_0^\infty \varPhi^{N-2}(p)e^{-p^2}\ dp. $$ In Rényi and Sulanke's paper Uber die konvexe ...
user311932's user avatar
4 votes
1 answer
96 views

Estimate of $\frac{\int x^{2p}\,e^{-x^{2n}\,+\,\omega(x,y)}\;dx}{\int e^{-x^{2n}\,+\,\omega(x,y)}\;dx}$

For every $x,y\in\mathbb R$ let $$ V(x,y) \,\equiv\, a\,x^{2n} + b\,y^{2m} - \omega(x,y)\,$$ where $a,b>0$, $n,m\in\mathbb N$, $n\geq m\geq1$, and $\omega$ is such that $\omega(x,y)/(x^{2n}+y^{2m})...
tituf's user avatar
  • 311
0 votes
1 answer
195 views

Sufficient conditions for finite mean of a non-negative random variable

Consider a continuous random variable that takes only non-negative values. Let the cumulative distribution function be $F(\cdot)$. Consider the following condition: $$\lim_{x\rightarrow\infty} x(1-F(x)...
liuchun deng's user avatar
0 votes
1 answer
86 views

Is integration against an indicator Wasserstein-Continuous

Let $\mathcal{P}_p(X)$ denote the Wasserstein space over a compact metric space $X$, and $1\leq p<\infty$. Fix a non-empty closed subset $C\subseteq X$. Then is the map: $$ \mathbb{P} \mapsto \...
ABIM's user avatar
  • 5,405
1 vote
0 answers
240 views

Riemann-Stieltjes integral of a distribution function

I recently learned the basics of Riemann-Stieltjes integral, and based on the sources I found, we can define the expectation of random variables quite naturally with the R-S integrals: if $X$ is a ...
gouhaha's user avatar
  • 21
0 votes
0 answers
72 views

Integration of fractional function over Rice distribution

Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as \begin{equation} \int_0^{\infty } {\frac{1}{{1 + {x^a}}}} f\left( {x|y} \right)\, dx \end{equation}...
hichem hb's user avatar
  • 377
8 votes
3 answers
628 views

Expected distance between two uniform points in distinct rectangles

Are there any good approximations (especially upper bounds) for the quantity $E(\|X_1-X_2\|$), where each $X_i$ is uniformly distributed in a rectangle $[a_i,b_i]\times[c_i,d_i]$? It does not appear ...
Tom Solberg's user avatar
  • 4,049
18 votes
0 answers
571 views

Fundamental Theorem of Algebra via multiple integrals

Consider the product of complex linear monic polynomials times polynomials of degree less than $n$, that is $\big( (z-\lambda), p(z)\big)\mapsto (z-\lambda)p(z)$. If we represent a polynomial by its ...
Pietro Majer's user avatar
  • 60.5k
1 vote
1 answer
853 views

Quadrature methods for high-dimensional Gaussian integration

Suppose that $f$ is the density of a high(-$d$)-dimensional Gaussian measure with mean $\mu$ and non-singular covariance matrix $\Sigma$. Let $g:\mathbb{R}^d\rightarrow \mathbb{R}$ be a continuous ...
ABIM's user avatar
  • 5,405
1 vote
1 answer
666 views

Definite integral of 2d Gaussian

Is there some analytic expression or even an approximation of the definite 2D Gaussian integral of the form: $$E=\int_a^b Dg \int_{cg+d}^\infty Dh$$ where $Dg=\frac{dg}{\sqrt{2 \pi}} e^{-g^2/2}$ and a,...
Uri Cohen's user avatar
  • 373
1 vote
0 answers
82 views

How should I proceed to solve this kind of integral equation?

Given $a>0$, $b>0$, I am trying to find the function $f_{a,b} : \mathbb{R}_+ \rightarrow \mathbb{R}_+$ such that for all $u \in \mathbb{R}_+$, $$\exp\left\{\;\int\limits_{\mathbb{R}_+} \ln\left(...
lrnv's user avatar
  • 686
1 vote
1 answer
493 views

Sufficient and necessary conditions for decomposing the sum of random variables

Given two $n$-tuple vectors $\vec{\alpha}=(\alpha_1,\cdots,\alpha_n)$ and $\vec{h}=(h_1,\cdots,h_n)$, where $h_i\ge0$, $\sum_{i=1}^nh_i=1$, and $\alpha_i\in(0,1)$, we consider a random variable $S$ on ...
RyanChan's user avatar
  • 550
6 votes
1 answer
343 views

Is there a standard way of defining the integral of an extended real function with respect to a finitely additive probability measure?

Let $X$ be a set, and let $\mu$ be a finitely additive probability measure defined on $2^X$. Let $\Phi$ be the set of functions from $X$ to $\mathbb R \cup \{-\infty, \infty\}$. Is there a standard ...
aduh's user avatar
  • 869