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.
9,027 questions
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More positive pivotal edges than negative ones at critical bond percolation on Z^2?
Consider critical bond percolation on $\mathbb{Z}^2$ inside a fixed rectangle $(0,0) - (an,n), a \geq 1$ and write $A$ for the event that there is an open crossing in the long (left to right) ...
- 960
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0
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265
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Expectation of a multivariate Gaussian over a plane
For a vector $X$ which follows a multinomial Gaussian distribution $N(\vec{0},\Sigma)$, a given vector $b$, and a known scalar value $c$, I would like to calculate the expectation :
$E[X|X^Tb = c]$
...
- 21
2
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1
answer
422
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Extending Wald's equation to two classes of i.d. random variables?
I try to adopt Wald's equation to a slightly more complex problem. In fact, after a full day, I found some solution now, but it has a confusing argument in the middle. Perhaps somebody can help me at ...
- 340
5
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1
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437
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Stationary, ergodic measures from the structuralist point of view
Stationary, ergodic measures are a class of objects very familiar to probabilists. In a sense, these are the weakest generalization of the classic case of independent, identically distributed random ...
- 8,512
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2
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2k
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random walk returning probability
Consider a two-dimensional random walk, but this time the probabilities are not 1/4, but some values p_1, p_2, p_3, p_4 with $\sum_{i=1}^4 p_i=1$. For example, from (0,0), it goes to (1,0) with p_1, ...
- 71
4
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2
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8k
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Upper bound on expectation value of the product of two random variables [closed]
Hello,
I am trying to find an upper bound on the expectation value of the product of two random variables.
So suppose x, y are two non-independent random variables, given that I know the distribution ...
- 343
8
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1
answer
527
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A q,t-extension of Plancherel Measure thru Yang-Mills Theory ?
Buried in the physics paper by Nekrasov and Okounkov, a strange identity is proven:
$$ \prod_{n > 0} (1 - q^n)^{\mu^2-1} = \sum_{\mathbf{k}} q^{|\mathbf{k}|} \prod_{\square \in k} \left( 1 - \frac{\...
- 22.8k
3
votes
2
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520
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Large deviations for missing mass
Let $\boldsymbol p=(p_1,p_2,\ldots)$ be a distribution over $\mathbb{N}$
and suppose that $S=(X_1,X_2,\ldots,X_n)$ are sampled iid according to $\boldsymbol p$. Define the
indicator variable $\xi_j$ ...
- 6,541
2
votes
2
answers
2k
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Total variation distance between a Poisson and a distribution with known mean/variance
Suppose that $\mu$ is the law of a Poisson distribution of mean 1, and that $\nu$ is the law of an unknown distribution on the non-negative integers, though I do know that its mean and variance are ...
- 163
6
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1
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658
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Calculating a specific joint probability involving sums of binomial distributions
The following might look like a simple problem - but the question has been unanswered for more than a week on math.stackexchange.com, and I have asked quite a few of the Ph.d. students at our ...
- 161
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1
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107
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Can one combine (join) probabilities from 2 aspects of a related process?
Consider 2 related aspects of a process for prices in a financial market:
time &
return.
Time
Say I've identified a distribution that reasonably models the occurrence of the lengths of price ...
- 111
1
vote
2
answers
724
views
Conditional probability with permutations
Hello,
This problem looks very simple and I conjecture it's true but I have a hard time proving it. It'd be very useful for my work (I'm doing a PhD) and I'll be glad to cite you in a future article ...
- 43
1
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0
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221
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Distance between probability amplitude functions
Suppose we have two probability measures $P_1$ and $P_2$ on some Riemannian manifold $(\Sigma,g)$. There are many potential distance measures between $P_1$ and $P_2$:
The Wasserstein distance
For $...
- 705
1
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2
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147
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limit of functionals on weak convergent random variables
Suppose real value random variables satisfy
$X_{n} \Rightarrow X$ (convergence in distribution)
as $n\to \infty$ in the same probability space
$(\Omega, \mathcal F, \mathbb P)$.
It is well known that ...
- 1,399
7
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0
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300
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Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
- 8,512
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135
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expected number of shared 1s between two binary strings from a given set
Let say, I have two binary strings with length N, chosen from a set where there are $2^N-K,(K \ge 0)$ independent strings. What would be the expected number of Ones at the same index from two randomly ...
- 21
4
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1
answer
904
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distribution of the number of empty bins in a multinomial setting
Let $(X_1,X_2,\ldots,X_k)$ be a multinomial random vector with parameters $n, p_1, p_2, \ldots, p_k$ (i.e., we throw randomly $n$ balls into $k$ bins, so that for each ball, the probability of landing ...
- 41
1
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0
answers
501
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Distribution of random vectors
Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$).
A vector $u\in ...
- 143
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4
answers
674
views
Brownian motion inside a domain
Consider a regular domain $D \subset \mathbb{R}^d$ and a Brownian motion $B_t$ conditioned to stay inside $D$ for time $t \in [0,T]$. In the limit $T \to \infty$ the conditioned Brownian motion ...
- 2,133
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1
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283
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Probability and events [closed]
Hi everyone
The question is the following:
A certain event may or may not take place. So we say that if we focus on it one time, it has a probability p of being satisfied (0 <= p < 1)
If we ...
- 3
9
votes
2
answers
441
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From very many sets of fixed measure in a probability space, can we select many that have a positive intersection?
I assume the following Lemma is either well known or, more probably, a
Corollary of a much stronger well known Theorem, and I would be grateful for a
reference:
For all $\delta\in (0,1)$ and all $\...
- 894
4
votes
0
answers
617
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Expected number of components with multiple cycles in a subgraph of a square lattice
Short version
Is there an understanding of the emergence and subsequent disappearance of components with zero, one, or more cycles in a random subgraph of a square or cubic lattice, as the edge-...
- 1,444
2
votes
2
answers
241
views
examples of random variable X independent to each of A and B, but not to (A,B) [closed]
What are some examples of random variables X, A, B such that X is independent to A, and to B, but not to A and B jointly, i.e., X is not independent to (A,B). In other words, $X \perp A$ and $X \perp ...
- 121
9
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2
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479
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Hyperreal finitely-additive measure on [0,1) assigning $b-a$ to $[a,b)$ or $(a,b]$ and infinitesimals to singletons
Is there a hyperreal-valued finitely additive measure on all the subsets of [0,1), or at least the Borel ones, that
assigns $b-a$ to $[a,b)$ and to $(a,b]$ for all $a\lt b,$ and
assigns an ...
- 2,555
2
votes
2
answers
230
views
Countably many random vectors and related problems.
Say I have a countably infinite number of iid random vectors $X_i:i\in\mathbb{N}$, each uniformly distributed on $[0,1]^k$ with say, $k=2$.
I need to evaluate stuff like:
$E_{X_0,X_1,\ldots}[\int_{[...
- 23
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votes
5
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6k
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Nonstandard analysis in probability theory
I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books:
Nelson (1987). Radically Elementary Probability Theory
...
- 1,302
7
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5
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974
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Collisions between rooks taking random flights on an N by M chessboard
I randomly place $k$ rooks on an (arbitrarily sized) $N$ by $M$ chessboard. Until only one rook remains, for each of $P$ time intervals we move the pieces as follows:
(1) We choose one of the $k$ ...
- 133
8
votes
2
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755
views
The Odds 3 (or More) Group Elements Commute
Some time ago I asked about the odds 2 group elements commute. I wonder about the odds that 3 group elements commute. Is there a "closed" formula for the sum
$$ \frac{1}{|G|^3} \sum_{g,h,k} \delta([...
- 22.8k
10
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2
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9k
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When do maximum and expectation commute?
Hi, I'm looking for conditions on $G(t,x)$ such that
$$
\sup\limits_{t\in [0,1]}E[G(t,X)]=E[\sup\limits_{t\in [0,1]}G(t,X)]
$$
where $X$ is a random variable (it's easy to see that $\sup\limits_{t\in [...
- 111
3
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1
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453
views
When can you describe a population and its component subpopulations with the same parametric family of distributions?
I believe that it is often the case that you are trying to select the best probability distribution to use to describe some phenomenon you are studying, and you have data not only for a population, ...
- 133
2
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1
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719
views
Lower bound on Bhattacharya distance between independent Gaussian distributions ?
I am interested in a lower bound on the Bhattacharya distance between two independent multivariate Gaussian distributions. To be precise, consider zero-mean independent Gaussian distributions $p_1\sim\...
- 163
4
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0
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233
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Convergence of probability measures on a generating field of a sigma-field
Let $(\Omega,\mathcal{B})$ be a measurable space and let $\mathcal{F}$ be a generating field of $\mathcal{B}$. Assume $\mathcal{F}$ is standard, i.e. it is countable, and any normalized, non-negative, ...
- 51
17
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1
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733
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Reference request: a conjecture of Rota on positive functions of a random variable
Rota and Shen's On the Combinatorics of Cumulants ends with a conjecture which I'll restate as follows:
Let $p \in \mathbb{R}[x_1, x_2, ...]$ be a polynomial such that, for any sequence $X_1, X_2, ...
- 118k
2
votes
2
answers
304
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Uniformly random planar map
Is there a way to sample a planar map uniformly at random? I am aware of the Cori-Vauquelin-Schaeffer bijection that can be used to sample and study uniformly random quadrangulations. There are other ...
- 1,989
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votes
3
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349
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minimum of two probability densities
Consider a smooth probability density $\pi(x)$ on $\mathbb{R}^d$. I am looking for natural for the integral $\iint_{u,v} \ \min\big(\pi(u), \pi(v) \big) \ du \ dv$ to be finite. If $\pi$ is a radially ...
- 2,133
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2
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462
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How to deal with this Chicken-And-Egg problem ?
Let's imagine designing an odds pattern for a game, in which players bet for win or lose.
Suppose the probablity of winning is $p$, thus the probablity of losing is $1-p$.
Now imagine $n_1$ people ...
- 99
60
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4
answers
3k
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Flipping coins on a budget
A coin is flipped $n$ times and you win if it comes up heads at least $k$ times. The coin is unusual in that you're allowed to pick the probability $p_i$ that it comes up heads on the $i$th flip, ...
- 82.7k
2
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2
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2k
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The probability distribution of random variable of random variable
In my understanding, random variable is a measurable function from a probability space to a measurable space. Suppose $X$ is a random variable from $(A, \sigma_{A},P_A)$ to $(B,\sigma_{B})$. And $Y$ ...
- 131
8
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2
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853
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A discrete random walk that avoids previously visited vertices for an exponentially distributed time interval
Imagine a discrete random walk on an infinite one-dimensional lattice where, for every unit interval of time, $(t_1, t_2, ...)$, the walker takes a step with uniform probability to its left or right. ...
15
votes
5
answers
921
views
What fraction of n x n invertible integer matrices contain at least one unit?
The question is simple:
What fraction of matrices in $G_n = \text{GL}_n(\mathbb{Z})$ have at least one unit entry (i.e., either $\lbrace\pm 1 \rbrace$)?
I'm not sure what the correct measure on $...
- 15.5k
10
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339
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Simultaneous strong law of large number classes?
Say that $C$ is a SSLLN class of subsets of some Polish space $V$ provided that for every sequence of Borel i.i.d.r.v.s $X_1,X_2,...$ with values in $V$, we almost surely have: For every $A$ in $C$, $\...
4
votes
3
answers
3k
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What is the name for a non-normalized distribution?
For some analysis work with probability distributions, I remember a common trick being to drop the "integrate to 1" requirement, so the set becomes closed under addition and is more convenient to work ...
2
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0
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228
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Invariant Measures of Markov Chains under Perturbations
This is a more specific version of a question I asked before without much luck. I believe this should be standard perturbation theory, but looking at Kato's book has not helped. Any references would ...
- 509
6
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0
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411
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Birth-Death Process associated with Orthogonal Polynomials
I have read in various places the following objects are related:
orthogonal polynomials
birth-death processes
Lattice paths
continued fractions
After a lot of searching online, I found sketches ...
- 22.8k
4
votes
1
answer
580
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Tracking down locality assumption in CHSH inequality
CHSH inequality requires both locality and realism. I will equate here realism with counterfactual definiteness.
Now counterfactual definiteness tells us that given two different measurements on the ...
0
votes
0
answers
111
views
Stationarity of an Integral Process
Let $f$ be a continous deterministic function defined on $\left[0,c\right]$ and $(B_{t}^{H})_{t\geq 0}$ be a fBM with $H\in \left(0,1\right)$. We define a Process $\left(X_{t}\right)_{t\geq 0}$ with
$$...
3
votes
1
answer
520
views
Results regarding $E[\min X,Y]$. when $X$ and $Y$ are independent, of given distributions.
Working on fairly unrelated stuff, I needed to prove the following, fairly easy results, and I wonder if anyone can provide references to the literature. Not being a probabilist I wouldn't know where ...
- 577
1
vote
1
answer
414
views
Fourier inversion formula for complex-valued random variables?
The characteristic function of a complex-valued random variable $X$ with pdf $\mu$ is given by
$$
\phi(t) = \int \exp[i \Re(\bar{t} X)] \; d\mu
$$
(or, so says Wikipedia). How does one recover the ...
1
vote
2
answers
1k
views
Representation theorem for continuous uniformly integrable martingales
For some time $u$ and positive continuous process $a_t$ adapted to $\mathcal{F}_t$ I have a (continuous-time) martingale defined as:
$$M_t(u) = \mathbb{E}[a_u | \mathcal{F}_t]$$
for $t\leq u$. I ...
- 667
5
votes
2
answers
472
views
Product of random diagonals on the unit circle
Let $P_1, P_2, ..., P_n$ be points randomly placed on a unit circle from a uniform distribution. Consider the product $D$ of all pairwise distances:
$D=\displaystyle \prod_{1\leq i < j \leq n} \...
- 413