All Questions
Tagged with pr.probability integration
133 questions
1
vote
1
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147
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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
$$...
- 13
1
vote
0
answers
302
views
Integration involving modified bessel function, exponential and power
I need to find the following integration.
$$
\int_0^a e^{-(N-1)x}(\sqrt{4x}K_{1}(\sqrt{4x}))^N
$$
where
$$
a>0, \quad N \geq 1
$$
Any help will be much appreciated.
BR
Frank
- 119
1
vote
0
answers
251
views
Inflated independent samples for Monte Carlo estimation
In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...
- 101
1
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0
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100
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Distribute Monte Carlo samples among dimensions
Simplified problem: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am ...
- 101
0
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1
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370
views
Closed-form CDF for bivariate normal distribution in point $(\Phi^{-1}(p),\,\Phi^{-1}(p))$
Let $\Phi(x)$ be a CDF of standard normal distribution and $\Phi^{-1}(p),\,p\in(0,1)$ its inverse.
It is evident that
$$
\mathbb{P}(X<\Phi^{-1}(p))=\Phi(\Phi^{-1}(p))=p,
$$
where $X\sim N(0,1)$.
Is ...
0
votes
1
answer
243
views
Integral form of expectation with respect to complex random variables [closed]
Let $h$ be a random variable and $g(h)$ be a real-valued function of $h$.
We know that if h is a real-random variable then:
$E_h[g(h)] = \int_{-\infty}^{\infty} f(h) g(h) dh$ where f(h) is the PDF of ...
- 13
0
votes
1
answer
195
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Sufficient conditions for finite mean of a non-negative random variable
Consider a continuous random variable that takes only non-negative values. Let the cumulative distribution function be $F(\cdot)$. Consider the following condition:
$$\lim_{x\rightarrow\infty} x(1-F(x)...
- 79
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 ...
- 143
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:
$$
\mathbb{P} \mapsto \...
- 5,405
0
votes
2
answers
246
views
Finding the expectation of $a \mathcal{Q} \left( \sqrt{b } \gamma \right) $, where $\gamma$ is a Gamma r.v
I'm trying to analytically find the following expectation
$$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right],$$
where $a$ and $b$ are constant values, $\mathcal{Q}$ is the ...
0
votes
1
answer
135
views
Bounds on expectation of $X/(X^2 + c)$ with $X$ ~ Gaussian and $c > 0$
I'm trying to compute expectation of $X / (X^2 + c)$ when $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, and $c$ is some positive constant. I think this cannot be solved ...
- 11
0
votes
1
answer
222
views
Behavior of the integral of products of probability densities
Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition
$$
T(x_1,\ldots,x_n) := \...
- 65
0
votes
1
answer
551
views
Surface of the cut of an ellipsoid / Marginal density of a multivariate normal over an affine space
So I'm trying to get the marginal density of a multivariate normal over an affine space
if $A$ is a matrix in $\mathbb{R}^p \times \mathbb{R}^n$ for $p < n$ and $B \in \mathbb{R}^n$, $\Sigma$ is a ...
- 1,902
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
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
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
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
$$
\dfrac{M_j}{...
- 57
0
votes
1
answer
102
views
Sign of expectation value
Consider a multivariate Gaussian-type measure $$d\lambda(x):=\nu_{\mu,\Sigma} e^{-\langle (x-\mu), \Sigma^{-1}(x-\mu) \rangle - \vert x \vert^2} $$
with vector $\mu \in \mathbb R^n$ and $\Sigma$ ...
- 536
0
votes
1
answer
503
views
Asymptotics of a 1D integral, or the orthant probability of an equicorrelated random Gaussian vector
Problem: Let $\phi(x)$ be the normal probability density function (pdf), and $\Phi(x)$ the normal cumulative distribution (cdf). I'm interested in the asymptotic behavior of the following integral
$I(...
- 968
0
votes
0
answers
36
views
Contribution of Fisher information near jump points in convolved probability distributions
I am trying to compute the contribution to the Fisher information from jump points $b_i(\theta)$ in the convolved function $f(x; \theta)$ with respect to the parameter $\theta$. I am unsure whether it ...
- 99
0
votes
0
answers
149
views
Reference book for a probability course
In the next months I am planning to deliver a (more-or-less) advanced course in probability theory. My students will have had already a first encounter with discrete probability theory (discrete ...
- 1,561
0
votes
1
answer
205
views
Inhomogeneous Markov chains and the product-integral as a solution to the Kolmogorov forward equation
We have a inhomogeneous continous $K$-State Markov chain $X(t)$ with transition intensity matrix $Q(t)$. Therefore its entries are:
$$q_{ij}(t)= \lim_{\delta \to 0} \frac{1}{\delta} \mathbb{P}(X(t+\...
- 3
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}...
- 377
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
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
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
0
votes
0
answers
453
views
Integral involving modified bessel function of second kind, exponential and power
I need to compute the following integral.
$$
\int_0^a e^{-bx}\sqrt{4(a-x)}K_1(\sqrt{4(a-x))}dx\,.
$$
where $$ a>0$$
and $b$ can be greater than zero or less than zero but it is not a complex ...
- 119
0
votes
0
answers
81
views
Why is $\mathcal{E}(X)=\mathcal{E}(X,X^*)$?
According to a course about $\sigma$-agebras in infinite dimensional space they said that it is easy to see that :
$$\mathcal{E}(X)=\mathcal{E}(X,X^*)$$
where:
$X$ is separable real Banach space.
$\...
- 121
0
votes
0
answers
454
views
Reference: Bochner Integral`
What would be an easily accessible book dealing with Bochner integration as applied to probability theory (I'm looking to understand random elements and their basic related concepts in a formal yet ...
- 5,405
0
votes
0
answers
232
views
Morphisms associated to measured spaces [duplicate]
In a previous discussion (von neumann algebras and measurable spaces), the connexion between von Neumann algebras and localized measured spaces was clarified. I would like to have a category theory ...
- 585
-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 \...
- 115
-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
-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