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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Expectation of a random sum
Let $X_1, X_2, X_3,\dots$ be an i.i.d. sequence of random variables with finite mean. Write $S_n=X_1+X_2+\dots+X_n$.
Let $N$ be a non-negative integer-valued random variable with finite mean. $N$ may ...
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Alternating colors on a line: infinitely often or converge?
Suppose we have intervals of alternating color on $\mathbb{R}$ (say, red / blue / red / blue / …). All intervals have independent length, with all red intervals distributed as $\mathbb{P}_{R}$, all ...
- 1,010
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Random walk on a simple finite network
Consider a graph $\Delta_N = \lgroup (x,y)\in\mathbb{Z}^2| x+y\leq N-1, x\geq 0,\ y\geq 0 \rgroup$ (set of edges is defined in a natural way): see here ).
Take a random walker that wonders around ...
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Card game / options pricing / Brownian bridge question
We play a game. I shuffle a deck of cards and start dealing them face up. After any card you can say "stop", at which point I pay you 1 dollar for every red card dealt and you pay me 1 for every black ...
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1
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mutual hitting measure between two sets
Given disjoint nonempty subsets $X_1, X_2$ of the state space of a finite irreducible Markov chain, there are unique measures $\mu_1$ on $X_1$ and $\mu_2$ on $X_2$ such that (a) starting from a $\mu_1$...
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Counting card distributions when cards are duplicated
If we have a deck of $48$ different cards and $4$ players each get $12$ cards, it is well known how to calculate the number of possible distributions: $\frac{48!}{12!12!12!12!}$
In a german card came (...
4
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2
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Supermartingales and convergence
These feel like basic enough questions, but I don't know where to find the answer.
Let $X_1,X_2,X_3,\dots$ be a supermartingale such that $|X_{n+1} - X_n| < K$ for all $n$ ($K$ fixed). Does the ...
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1
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Can you interpret this divergent integral?
In this ArXiv paper by Wilk and Wlodarczyk (published in Physical Review Letters), equation 16 has essentially the following definition of a function:
$$\text{f(x)=}\frac{c}{2Dx^2}\exp[\int^x_0 \frac{\...
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5
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Coin flipping and a recurrence relation
How can one solve the following recurrence relation?
$f(n) = 1 + \frac{1}{2^n} \sum_{k = 0}^n {{n}\choose{k}} f(k)$
$f(0) = 0$
As it happens, I can show $f(n) = \Theta(\log n)$ through other means (...
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1
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A random walk on natural numbers
We are taking a random walk on the set of natural numbers. If we are at $M$, then with probability 1/4, we stay at $M$, with probability 5/12 we move to some random number less than or equal to $M/2$, ...
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A "random variable" with infinite value
A random variable (r.v.) is a (measurable) fucntion from probability space $\Omega$ to $\mathbb{R}$. In our applied problem, the best model would be an extended "r.v." from $\Omega$ to $\mathbb{R}\cup\...
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finding numbers at k hamming distance
Guys,
I have N < 2^n randomly generated n-bit numbers stored in a file the lookup for which is expensive. Given a number Y, I have to search for a number in the file that is at most k hamming dist....
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Stable distributions for Lindeberg exchange strategy?
Terence Tao has mentioned the importance of the Lindeberg exchange strategy, citing as an application how it was used in the proofs of some recent results relating to universality laws for random ...
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On random Dirichlet distributions
Fix a dimension $d\ge2$.
Let $Q_d$ denote the positive quadrant of $\mathbb{R}^d$, that is, $Q_d$ is the set of points $\mathbf{x}=(x_i)_i$ in $\mathbb{R}^d$ such that $x_i>0$ for every $i$.
For ...
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Martingales in both discrete and continuous setting
I am wondering, polynomials like
$S_n^4-6n S_n^2+3n^2+2n$ for $$S_n=\sum_{i=1}^n{X_i}$$ where $$\mathbb{P}(X_i=1)=\mathbb{P}(X_i=-1)=\frac{1}{2}$$ is a martingale (under the conventional filtration). ...
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probability question regarding brownian motion
I am wondering where to start with questions like:
Given a BM $dX_t=\mu t+\sigma dB_t$, having started at $X_0=0$. What is the probability that $X_t$ does not hit 0 in the time interval $[a,T]$ where ...
- 255
6
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1
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Why is the dimension of Gaussian variables is bounded by the dimension of the space?
I'm looking at a probabilistic proof of a local version of Dvoretzky's theorem in Pisier's manuscript "Probabilistic Methods in the Geometry of Banach Spaces."
For each $\epsilon >0$ there is a ...
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1
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1
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An infinite Gaussian mixture with mixing parameters being also Gaussian
A finite Gaussian mixture with $k$ components has a probability distribution function $p(y|\mu_1,...,\mu_k, \sigma_1, ..., \sigma_k, \pi_1, ..., \pi_k)=\sum_{j=1}^{k} \pi_j\mathcal{N}(\mu_j, \sigma_j^...
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3
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Random knot on six vertices
This question is inspired by Joseph O'Rourke's beautiful question on random knots. Choose an random ordered 6-tuple of points on the unit sphere in $\mathbf{R}^3$, and form a knot by connecting ...
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1
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1
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Probability of d distinct outcomes after n trials
Hi,
I'm trying to find the probability that after n trials of a multinomial rv, there have been exactly d distinct outcomes.
What I'm ultimately trying to calculate is the expected number of trials ...
- 13
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How I determine the probability that an unknown probability value is greater than others in a set?
I have a number of known beta distributions for different unknown probability values.
Given the beta distributions, I want to determine the probability that each specific unknown probability values ...
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Concentration of measure and bounds on variance
I am trying to characterize the sensitivity of a function $f: R^N\to{}R$ to the perturbations in the input vector $\mathbb{x}=\left[x_1,\dots{}x_N\right]$. For that purpose, I evaluate Cramer-Rao ...
4
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Balls-and-bins type problem
Suppose I have an n-by-n array of bins, and I want to choose k (k >= n) bins from these n^2 bins such that each row of the array has at least one bin chosen. How many ways are there of doing this?
...
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Borel-Cantelli Lemma on MCs (absorbing states)
hi, I'm sorry if the question is silly, but I couldn't get my head around it for a while now.
In Markov Chains (MC) proving that a state is either recurrent or transient is through Borel-Cantelli ...
- 355
4
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1
answer
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Intersection Probabilities for Random Walk in d>2
I'd like to get asymptotics on the probability that n independent random walks coalesce. Start with n independent walks. As soon as two walks intersect they become one walk and continue evolving as ...
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2
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Weighted Poincaré inequality
Consider a probability distribution $\pi$ with density $e^{-H(x)}$ on $\mathbb{R}$. Let us say that there is a Poincaré inequality with weight $w$ if for any smooth function $\phi$ satisfying $\int \...
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Prove: if a1,...,an are uniformly distributed unit vectors, then a1*a1'+...+an*an'=n/2*I
Hello everyone,
I have a very interesting question on orthogonal projection matrices. Intuitively it is quite straightforward to understand. But for me it is not easy to prove.
In $R^2$ space, $a_i$,...
3
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2
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The space of probability measures and its intersection with hyperplanes in the space of measures
Let $X$ be some uncountable standard Borel space (e.g., the real line).
Let $D$ be the set of Borel probability measures on $X$.
Let $M$ be the set of signed Borel measures on $X$
Now let $p_1,...,p_N$...
user12713
4
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1
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Self Avoiding Walk Pair Correlation
Let $\gamma(i)$ be a self avoiding walk (SAW) on a 2D lattice $L$ (a square lattice for example) starting at a predefined origin ( $\gamma(0)=(0,0)$ ) and having length $n:=\ell(\gamma)$. Furthermore, ...
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when does inner product with fixed vectors determine joint distribution?
Given a random vector $(X_1,X_2)$. If $aX_1 + bX_2$ is Gaussian for all pairs $a,b$, then $(X_1,X_2)$ is jointly normal. More generally, is the following statement true?
If $aX_1 + bX_2$ has the same ...
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Modeling question: how often does "the world's oldest person" die?
This story yesterday (no need to follow the link to understand the question!)
http://www.cnn.com/2011/US/02/01/texas.oldest.person.dies/index.html?hpt=T2
reminds me that I've often wondered about ...
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Is there an f(x) such that P[f(a) >= f(b)] = a/(a+b) given a set of possible values for a and b?
I need a function $f(x)$ such that given a set of values $X=\{x1, x2, ...\}$ and a function $rand()$ that returns a value between 0 and 1, $f(x)$ returns a value that increases with x such that the ...
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deriving angular central gaussian distribution from a multivariate normal distribution
The angular central Gaussian (ACG) distribution on $(p-1)$-dimensional sphere $\mathbb{S}^{p-1}$ for a symmetric positive definite parameter matrix $\mathbf{A}$ is defined as
$$f(\mathbf{x},\mathbf{A}...
23
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2
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Random permutations of Z_n
In "The maximum number of Hamiltonian paths in tournaments" by Noga Alon, the author states the following without proof (equation 3.1):
"Consider a random permutation $\pi$ of $\mathbb{...
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Independent Events Inducing Probability Measures
Let $\mathcal{F}$ be a sigma algebra over $\Omega$ and $M$ the set of all probability measures on $\mathcal{F}$. Let $\mathcal{C}$ be some collection of pairs $(A,B)$ with $ \ A,B\in\mathcal{F}$. Now ...
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Does there exist an event independent of a given sigma-algebra?
The following question came up in a discussion with my advisor:
Let $(\Omega, \mathcal F, \mathbb P)$ be a non-trivial probability space, and suppose that $\mathcal G$ is a proper sub-$\sigma$-...
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Is there any straightforward way to substitute for Gaussian/Brownian assumptions in financial mathematics?
A huge amount of financial mathematics assumes Gaussian distributions of risks and Brownian movement of prices. What efforts have there been to replace these with heavy-tailed distributions? For ...
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Exact simulation of SDE
Consider a one dimensional SDE of the form $dX_t = \mu(X_t) dt + \sigma dW_t$, where $\sigma>0$ is a constant. Under mild regularity assumptions on $\mu(\cdot)$, one can exactly simulate ...
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Time-integral of a smooth, vector-valued function of a planar Brownian bridge
I'm looking for information on how to compute the distribution of the random vector
$$Z = \int_0^t f(B_s) ds$$
where $t>0$ is fixed, $B_s$ is a 2D Brownian bridge with $B_0 = 0$, $B_t=b \in \...
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Radon transform and Log-concavity
This question is related to (but different from) that of Darsh Ranjan.
Is there a characterization of the functions $f:\mathbb R^n\rightarrow\mathbb R_{\ge0}$ whose Radon transform $\hat f(\omega,t)$...
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Follow up question, Ornstein-Uhlenbeck Extension with n mean-reversion values
Hi this Question follows after the answer of Douglas Zare to this post :
So let's be given a positive function $V(x)$ which is smooth enough to have Lispchitzian derivative at all point. Moreover ...
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An $L^0$ Khintchine inequality
Suppose that $\epsilon_1,\epsilon_2,\ldots$ are IID random variables with the Bernoulli distribution $\mathbb{P}(\epsilon_n=\pm1)=1/2$, and $a_1,a_2,\ldots$ is a real sequence with $\sum_na_n^2=1$. ...
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Anti-concentration of Bernoulli sums
Let $a_1,\ldots,a_n$ be real numbers such that $\sum_i a_i^2 =1$ and let $X_1,\ldots,X_n$ be independent, uniformly distributed, Bernoulli $\pm 1$ random variables. Define the random variable
$S:= \...
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Maximum of a set of sums of iid random variables
Consider some probability distribution $D$ over non-negative reals with finite expectation $\mu$. Now for any positive $T$ consider sums of $T$ iid random variables drawn from $D$. A single sum of ...
- 501
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4
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What would be a fractional Poisson Process like
I think that the definition of fractional Brownian Motion is widely known (for example as a Gaussian Process with particular variance covariance stucture parametrized by the so-called Hurst index).
...
- 1,334
8
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6
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Diffusion sample paths as deformed Brownian sample paths
Suppose $X$ is a non-explosive diffusion with dynamics
$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$,
where $W$ is a standard Brownian motion. My intuition about $X$ is that if $\mu$ and $\sigma$ are ...
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1
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Convergence of sets
Let $E$ be a compact subset of $\mathbb{R}^n$. Let the density function $\phi(x,y)$ be Lipschitz continuous and such that
$$
\int\limits_E \phi(x,y)dy=1
$$
for all $x\in E$. Let us consider the non-...
- 1,655
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0
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484
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Stable local limit theorems
Consider a sequence of integer valued indentically distributed centered independent random variables $X_1, X_2, \ldots$ with the additional condition that the support of $X_1$ is aperiodic. Suppose ...
4
votes
1
answer
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Square of Binomial Coefficient
Background
I'm modeling Genetic Algorithm(GA) with Markov chains and deriving the expression for the expectation of the first hittig time in the MC with 1 absorbing state and $l-1$ transient states. ...
- 355
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4
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Reference request: probability / ergodic theory without measure spaces
In his notes on free probability, Terence Tao describes a general approach to non-commutative probability which prioritizes the algebra of random variables above the sample space; I find this ...
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