Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
9,022 questions
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integral of a squared Bessel process
Let $X_t$ be a Squared Bessel Process ($BESQ$). Define $Y_t=∫_0^tX_sds$. Do we know whether $\lim_{t→∞}Y_t=∫_0^∞ X_sds$ is finite or infinite? Does it depend on $BESQ$ parameter?
Edit.
It is obvious ...
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660
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Tail bound for maximum of independent (but not identical) binomial random variables
This post derived the tail bound for the maximum of independent and identically distributed binomial r.v.'s based on normal approximation. Is there a similar result in the literature for finding the ...
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201
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The distribution of the maximum of a series of extreme value type I random variable
I have an infinite series of independent identically distributed random variables $\{X_i\}_{i=1}^\infty$ which follows extreme value type I distribution which can be found [here] (https://en.wikipedia....
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Bounded convergence for expectation of random variables [closed]
I have a random variable $X$ defined on $(0,\infty)$. For each $n\in \mathbb N$,
define $X_n = X \mathbf{1}_{0 < X \leq C_n}$, where $C_n$ is a monotonically increasing sequence of positive numbers ...
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How to compute the clustering coefficient of a random graph?
How is the clustering coefficient defined for random graphs? For example, a first definition could be calling clustering coefficient of a random graph the expected value of the clustering coefficient ...
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502
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Does the conditional expectation of a measurable process always have a progressive measurable version?
Does the conditional expectation of a measurable process always have a progressive measurable version? For example, X_t is a measurable process, but not progressive measurable, let Y_t=E[X_t|F_t], ...
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151
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Can an unskewed distribution be expressed as product of a normal and another distribution?
Let $x$ be a continuous random variable with zero mean and zero skew. What are the conditions under which we can say that $x$ can be expressed as the product $z y$ where $z$ is a standard normal and $...
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905
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Parameter estimation distribution for hypergeometric distribution
Let the hypergeometric distribution is given by $h(k\mid N;M;n)$, where
$k$ is the number of observed successes,
$N$ is the population size,
$M$ is the number of success states in the population and
$...
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198
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The eigenfunctions of an operator commuting with all rotations.
When reading the paper
E. Carlen, J. Geronimo & M. Loss: SIAM J. MATH. ANAL., vol. 40, no. 1, 327-374
I found an argument like the following.
Given an bounded and self-adjoint linear operator ...
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503
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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(...
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Solution for 2 variable recurrence for a problem similar to gambler's ruin [closed]
Let $p_A,p_B$ with $p_A+p_B=1$, $p_A \geq p_B$ be the probabilities of a biased coin flip. Player $A$ gets 1 point if the coin flip gives heads, $B$ gets 1 if tails. The player who reaches $N$ points ...
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143
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triply line stochastic matrix with maximum total on some cubes
A triply line stochastic matrix (t.l.s.m.) of size $N$ is a 3-dimensional array $(a_{ijk})_{i,j,k=1}^N$ with nonnegative entries, whose any row, column or line sums up to $1$. Their set is $TLS_N$.
...
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Question on Palm distribution in Kallenberg's book
I have a question regarding Kallenberg's "Foundations of Modern Probability" 2nd ed, a statement right after Lemma 11.2: "...$\eta\{0\}=1$ a.s.".
Let me state the setup:
$\xi$: a random measure on $\...
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Limit of iterative addition of a mean-preserving spread
Suppose I iteratively add a given mean-preserving spread to a random variable. In the limit, will exactly half the mass be above $0$?
Formally: Let $X$ be a random variable, and let $\varepsilon_1,\...
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414
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Exact formula for computing n-step transition probability of random walks with self-transitions
Consider a semi-infinite random walks $X_n$, $n=0,1,2,\ldots$, whose state space is a set of consecutive integers and whose one-step transition probabilities are $P_{ij}=\mathrm{Pr}\{X_{n+1}=j|X_n=i\}$...
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217
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Reproducing Kernel Hilbert Spaces with positive kernels
In my research I'm dealing with the following question.
Let $E$ set, $K:E \times E \to \mathbb R$ a positive type function, and $\mathcal H := \mathcal H(1+K)$ (in the sense of the Moore theorem). ...
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239
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Transition probabilities for the symmetric random walk on the integers
I found that most references for the symmetric random walk on the integers are for the discrete time case, i.e. the ones that gives us explicit transition probabilities. Now, I am looking at a random ...
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286
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Are $\left[\begin{matrix}x_\ell \\ x_\ell\varphi_k^\ell\end{matrix}\right]$ linearly independent?
Let $\varphi_k\in\mathbb{C}$ be a primitive $k$-th root of unity, and define the sets
$$S_\ell:=\left\{\left[\begin{matrix}x\\x\varphi_k^\ell\end{matrix}\right]\in\mathbb{C}^{2n}\;\middle|\;x\in\...
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Local extrema of a posterior probability
Let $x$ be a binary random variable and $z$ be an arbitrary random variable. $x$ and $z$ are, in general, not independent.
Let $y_1, \ldots y_n$ be $n$ identically distributed binary random variables ...
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204
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Alternative formula of a Green's function for average density of eigenvalues of random matrix
A Green's function is defined as follows:
$$G(\omega) = \frac{1}{N}\mathrm{E}\big[ \mathrm{Tr}\frac{1}{I\omega - J} \big]$$, where $I$ is the $N$-dimensional identity and $E$ means expectation value ...
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152
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Feller processes / probability generators
I am looking for a example of a function in $C_0(\mathbb{R})$ such that $f',f'' \,\text{and}\, f''' \in C_0(\mathbb{R})$ with
$$ \inf f < \inf (f-a*f''')$$ for some $a>0$, but I couldn't find ...
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76
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A way to possibly calculate one Binomial CDF function from another closely related one?
Let $y < z$ be two numbers between $0$ and $1$, is there a way to relate the CDF functions $F_{n,y}(s)$ and $F_{n,z}(s)$... or approximate one from another, without just saying $F_{n,z}(s) \le F_{n,...
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708
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Distribution of bounded summation of i.i.d random variables
We have a set of positive random variables $\boldsymbol X=\{X_1, X_2,\ldots\}$, where $X_1, X_2,\ldots$, are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for $X_i$ ...
- 143
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374
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Approximation of general measurable maps by simple functions [closed]
Let $f : (\Omega, \mathcal F) \to (\mathbb R, \mathcal B(\mathbb R)$ be a measurable map, then it is well-known that $f$ could be approximated by a sequence $(f_n)$ of simple measurable functions, ...
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Definition of "Expected/Unexpected Event"
Background of my question is Martin Gardner's "unexpected hanging" paradoxon, which has once again be the subject of an article in a popular-scientific magazin (this time because this year it has been ...
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probability in galton watson processes [closed]
I am trying to study the Elementary new proofs of classical limit theorems for Galton Watson processes written by Jochen Geiger.
I don't understand what Z_(n,i) stand for.
And in the proof of Theorem ...
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2
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627
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Generalized expression for balls and bins problem
$n$ number of balls are thrown randomly to $m$ number of bins, standing in a row. The balls are labeled as $1,2,3,....n$ and bins are also labeled as $1,2,3,...,m$. The probability of $i_{th}$ ball ...
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184
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Norm of matrix with randomly deleted entries
Let $A$ be an $n \times n$ matrix with real entries and let $B$ be the random matrix whose $(i,j)$ entry is $$B_{i,j}=v_{i,j}A_{i,j}$$ where the $v_{i,j}$ are i.i.d Bernoulli random variables with $P(...
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292
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Volume of randomly changing sphere follows beta distribution
We are given $X,X_1,\ldots,X_N$ independent and identically distributed $k$-dimensional vectors. For a given query point $X_q\in\mathbb{R}^k$ assume without loss of generality that $X_1,\ldots,X_m$ ...
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124
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Stationary distribution of random walk alias solving uncountably many linear equations [closed]
Let us have interval $I = (i_1,i_2)$, function $f_1 : I \mapsto I$, function $f_2 : I \mapsto I$.
Let $x_0$, $x_1$, $x_2$, ... be series of random variables from interval $I$ denoting random walk. ...
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What are some examples of isotrophic sets?
What are some examples of isotrophic sets? and is there a "good" way to describe them?
Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all $...
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Nagakami behavoir
Is the sum of square Nagakami random variables Erlang distributed?
What is the distribution of euclidean norm of complex Nagakami?
Cheers!
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Monotonicity of the gap of permutated sequence
Let $a$ be an arbitrary sequence and denote by $\mbox{gap}_k(a) = a_{(k)} - a_{(k+1)}$, where $a_{(k)}$ is the $k$th largest component of $a$. Of course, $k+1$ should be no larger than the length of $...
- 773
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329
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Measurable functions lifted onto a space of point measures are measurable
I've been reading [1] and attempting to prove statements given without proof. In the paper the authors construct a measurable space of measures over a base space, and as an aside show an elegant way ...
- 273
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227
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two correlated processes
I apologize if this question is not placed in the right place. But I am having a hard time to figure it out. It would be greatly appreciated if some one could help me out.
Assume that there are two ...
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324
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On the superior of generalized Ornstein-Uhlenbeck process
Let us consider a generalized O-U process $X_t \in L^2[0, 1]$ defined by the following spde:
$dX_t = \frac{1}{2}\partial_x^2X_t + dW_t, $
$\partial_x X_t(0) = \partial_x X_t(1) = 0, $
$X_0 = 0, $
...
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204
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Expected number of samples above certain value of a normally distributed variable with a given sample mean
Suppose $n$ values, $X_1,...,X_n,$ are generated by a random number generator with normal distribution $N(0,1).$ Suppose that the (sample) mean of $X_1,...,X_n$ is $\mu.$ What is known about the order ...
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467
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On the expected value of a random integral:
Is it possible to find the expected value of $u(t)$ in terms of the following information:
$$u(t)=\int_{0}^{t}(t-s)(f(s)+(T-s)Y)X_sds$$
where:
$X_s$ is a wide sense stationary process with known ...
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4k
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Convergence of the empirical distribution function
Let $\alpha\in\mathbb R^d$, with $\alpha\neq 0$. Take a sequence of iid random variables $X^{1},\dots,X^{n}$ with values in $\mathbb R^d$ and denote $R$ the cdf of $\alpha X^1$ (where the product is a ...
- 293
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Null space of random $(0,1)$ binary matrix [closed]
What can be said about the null space of random $(0,1)$ rectangular binary matrices? In particular, I am interested in the probability that there is any non-zero vector with only integer coordinates ...
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162
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Extracting moments from a special Z-transform
Suppose I have a sequence of positive continuous random variables $\{X_k\}_{k=1}^\infty$ with (unknown) MGF's $M_{X_k}(s)$. Furthermore, it is known that
\begin{equation}\frac{X_n-n\mu}{\sqrt{n}\sigma}...
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161
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Ratios of random variables with weak moment condition
Let $X_n$ be a sequence of iid positive random variables. Assume that $X_n$ has finite $\alpha$th moment for some value $\alpha \in (0,1)$, but infinite first moment. Assume also that the reciprocal $...
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379
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Continuity of caglad process
Consider a non increasing, caglad process $(X_t)_{t\geq0}$ such that, for each $t$, the distribution function $F_t(x):=P(X_t\leq x)$ is a continuous function of (real) $x$. Are there any sufficient ...
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733
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$L^2$ convergence of a tight sequence [closed]
Let $(X_n,n\geqslant 1)$ be a tight sequence of stochastic processes defined on the same probability. Suppose $\lVert X_n\rVert_{L^2}$ converges to $\lVert X\rVert_{L^2}$. Under what conditions do we ...
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2k
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Markov Chain: state reduction
Hi I am trying to understand a proof in a paper (written by Isaac Sonin), I don't know if anyone could give me a clarification on the following:
Firstly we have a Markov chain $\{Y_k\}$ with finite ...
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514
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Relating percentiles to moments [closed]
There are at least two ways people look at statistical data:
A. For mathematicians, scientists, engineers, economists and such the most familiar distribution parameters would be analytical: mean, ...
- 2,205
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4k
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Calculate the probability of winning for a selected tic-tac-toe player
I am not a mathematician, I am a programmer. Sorry, if formulation of the problem is inexact.
I want to calculate the probability of winning for a selected tic-tac-toe player.
I have a directed graph ...
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1
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89
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Is a $CD(K,\infty)$ space a length space?
Let $(X,d)$ be a complete and separable metric space endowed with a nonnegative Borel measure $\mu$ with support $X$ and satisfying
\begin{eqnarray}
\mu(B(x,r))<\infty,\quad\mbox{for every }x\in X\...
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197
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Area on the unit sphere swept out by big circles corresponding to a curve
For a point on the unit sphere, we know the plane perpendicular to the line through the origin and the point cuts the sphere with a big circle. When the point moves along a sphere curve, the ...
0
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1
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145
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Limiting probability question [closed]
Let $X$ denote an $m\times n$ matrix and suppose that each value $x_{ij}$ is an integer that is selected uniformly at random from ${1,\dots,n}$, independently of all other values. If we fix $m$ and ...
