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.
9,023 questions
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Is pi a good random number generator?
Part of what I do is study typical behavior of large combinatorial structures by looking at pseudorandom instances. But many commercially available pseudorandom number generators have known defects, ...
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4
votes
2
answers
13k
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how to find derivative of a stochastic process?
Consider the following equation for $X(t)$:
$$X(t)=e^{-bt}X(0)+\sigma\int_{0}^{b}e^{-b(t-s)}dW(t) \, ,$$
where $0 < b, \sigma\in\mathbb{R} $, $X(0)$ is the initial distribution of $X(t)$, ...
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1
answer
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Shuffling decks of cards where not all cards are distinguishable
Suppose a deck of cards consists of $a_1+a_2+\cdots+a_k$ cards of $k$ types, where there are $a_i$ indistinguishable cards of each type. How many shuffles does it take, on average, to randomize the ...
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probabilities of increasing events under different product measures.
Let $(B_i)$ be a collection of i.i.d. random variables taking values 0 or 1.
Suppose $0 < x^- < x^+ < 1$. Consider two different "success probabilities" for the i.i.d. collection $B_i$: ...
- 3,937
1
vote
1
answer
980
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Maximum of Convex combination of random variables
Let $X,Y$ be two independent random normal standard distributions. Consider a function $u(x)=\sqrt[ ]{x}$ if $x\geq{}0$ and $u(x)=-2\sqrt[ ]{-x}$ if $x<0$. Define $Z=aX+(1-a)Y$.
Question: How do ...
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0
votes
1
answer
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Conditional Covariance
It is well known that for two increasing functions $f$,$g$ and for any random variable $X$ then $cov(f(X),g(X))\geq{}0$. Now assume $f,g$ have the same domain $D$ and let $A\subset{}D$. What can I say ...
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2
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Failure probability formula
In page 21 of A Problem seminar, D. J. Newman presents a novel way (at least for me) to determine the expectation of a discrete random variable. He refers to this expression as the failure probability ...
- 8,842
16
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2
answers
14k
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Derivative of a random variable
Hi,
If I have two i.i.d random variables $X,Y$ and a parameter $a$. If I define a new random variable $Z(a)=aX+(1-a)Y$.
Does it makes sense to talk about first, second derivative of the random ...
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3
answers
801
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Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance Part II
For some context see Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance
As per Noah's answer and JBL's comment this was false as stated. However, I think the following ...
9
votes
1
answer
526
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Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance
Problem: Consider a random walk on the lattice $\mathbb{Z}^2$ where on each iteration a particle either stays at its current location or moves to a neighboring vertex with probability 1/5. We start ...
1
vote
1
answer
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A question about Chapter 12 (Vapnik-Chervonenkis Theory) of 'A Probabilistic Theory of Pattern Recognition'
Hi,
Can anyone familiar with the book 'A Probabilistic Theory of Pattern
Recognition' or the theory described help me out?
See quote from chapter 12, 'Vapnik-Chervonenkis Theory', of 'A
...
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votes
3
answers
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Some models for random graphs that I am curious about
G(n,p)
We are familiar with the standard notion of random graphs where you fixed the number n of vertices and choose every edge to belong to the graph with probability 1/2 (or p) independently. This ...
- 24.7k
2
votes
3
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2k
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Föllmer: "Calcul d'Ito sans probabilités" in English or German?
Does anybody know a translation of Föllmer: Calcul d'Ito sans probabilités in English or German?
It seems to be a very interesting text - Abstract: "It is shown that if a deterministic continuous ...
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6
votes
2
answers
428
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how to sample a conditioned diffusion
there are several reasons why we could be interested in sampling conditioned diffusions:
if we observed a diffusion at discrete time and want to do some kind of inference on the parameters of the ...
- 2,133
10
votes
3
answers
1k
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Random walks and Lyapunov exponents
Given a sequence $Y_1, Y_2, \dots$ of i.i.d. matrices in $\mathrm{GL}_n(\mathbb R)$, there is a theorem of Furstenberg and Kesten which says that if $\mathbb E(\log\|Y_1\|)$ is finite, there exists a ...
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2
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probability of subset sum after rolling dice 4 times [closed]
If we roll 4 dices (fair), what is the probability of "sum of subset" being 5. e.g. 1432,1121, 2344, 2354 have a subset sum of 5. Can you illustrate how to calculate this.
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0
answers
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Constructive aspects of Caratheodory's theorem in convex analysis
Let me paraphrase Caratheodory's theorem in a probabilistic setup:
Let $X$ be a real-valued random variable. For $k = 1, \ldots, m$, let $f_k: \mathbb{R} \to \mathbb{R}$ be a continuous function such ...
- 1,839
8
votes
1
answer
856
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tetrahedron edges probability
If 6 numbers are chosen at random, uniformly and independently, from the interval [0,1], what is the probability that they are the lengths of the edges of a tetrahedron?
I wrote some code and ...
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2
votes
1
answer
2k
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Stationary Solutions of stochastic differential equations
When does the stationary density of an homogeneous Markov process exist?
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1
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The mathematical theory of Feynman integrals
It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...
- 23.5k
-3
votes
1
answer
440
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Conditional expectation [closed]
Given E[v|X=x]=g[x] and the pdf of X (f[x]), how to calculate E[v|x>=x0]? The pdf of V or the joint pdf of V,X are unknown. My guess is that this problem has no solution.
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2
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Another question on Øksendal's book
Hi
On page 98 "Stochastic differential equations" of Øksendal, 6th edition,
the author writes that $$\int_{0}^{u}\Big(\int_{0}^{t}\frac{\partial}{\partial t}f(s,t)dR_{s}\Big)dt=\int_{0}^{u}\Big(\...
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1
answer
3k
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Comparing normally distributed variables
Given two normally distributed variables x_1, x_2, is there a non-simulation method of calculating the probability that ...
- 23
4
votes
1
answer
985
views
weak convergence in infinite dimensional spaces
Weak convergence can be tricky when dealing with infinite dimensional spaces. For example, the usual Levy's continuity theorem does not extend readily to separable Banach spaces.
Consider a (...
- 2,133
3
votes
1
answer
704
views
Expectation Maximum
Hi,
In my research, I have the following problem. Let $X,Y$ two i.i.d random variables and a function $u(x)=x^2$ if $x>0$
and $ u(x)= -\beta\ (-x)^2$ if $x\leq{}0$ with $\beta\geq{}1$
I need to ...
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votes
1
answer
251
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what kind of probability distribution can be used to model numeral-noun combinations?
I'am learning German (any other language will do).
I choose randomly a countable noun - for example the noun "Hotel".
I write the noun into google together with German numeral for "two" in double ...
- 173
4
votes
1
answer
383
views
initial condition of a diffusion approximation
I am trying to prove that a certain sequence of Markov chains $x^N_k$ converges towards a diffusion process. The invariant measure of $x^N$ is $\pi^N$ and the Markov chain $x^N$ is started in ...
- 2,133
6
votes
1
answer
323
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Geneology of survivors in a critical discrete Galton-Watson process
Hello. After flipping through a few textbooks on birth-death processes, I can't seem to find anything about genealogical distribution of survivors (conditioned on non-extinction). What I am looking ...
- 61
1
vote
3
answers
312
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Chance of something being fixed [closed]
I'm fixing a software defect that occurs 1 in n test runs. If I want to know that the probability of it being fixed is >= p for some 0 <= p < 1, how many times, m, do I need to run the test ...
- 43
4
votes
0
answers
167
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The mathematics of Schellings segregation model
For those who don't know the model. You can read this pdf. I want to find what is the probability that 2 nodes are each others neighbors when the algorithm converges (i.e. when all nodes are happy).
...
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2
votes
2
answers
5k
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Risk dice roll probabilities [closed]
Hi,
I'm wondering what probablity can tell me about dice rolling strategies in the game Risk. When a player attacks another player, they can roll up to 3 dice. The defending player can choose to ...
- 121
15
votes
2
answers
1k
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self-avoidance time of random walk
How many steps on average does a simple random walk in the plane take before it visits a vertex it's visited before?
If an exact formula does not exist (as seems likely), then I'm interested in good ...
- 19.7k
7
votes
1
answer
296
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Existence of Limiting Distribution for Moving Regions in Stat. Phys. Models
As the title (hopefully) suggests, I've been trying to prove (or disprove) the existence of a limiting distribution for a certain projection in a statistical physics model. I'll give the details of ...
- 101
1
vote
1
answer
22k
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Covariance and standard deviation relationship
I would like to know if an increase in the covariance between two variables would imply that the standard deviation for one of the variables has increased?
This is assuming that the standard ...
- 21
4
votes
1
answer
938
views
Random projection and finite fields
Suppose we have, say, $n$ $2n$-dimensional linearly independent vectors over $\mathbb{F}_2$. We do a projection on a random $d$-dimensional subspace. We are interested in probability that images of ...
- 1,791
6
votes
1
answer
816
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edge distribution of random Young's tableaux from Okounkov's "random matrices and random permutations"
I am reading the paper "random matrices and random permutations" by Andrei Okounkov, which is very beautifully written. I just have several technical questions about some of the computations:
1. in ...
- 4,466
1
vote
1
answer
787
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Homology of algebraic varieties in Okounkov's paper on enumerating algebraic curves
This is a series of questions in chronological order. I am lately trying to understand Okounkov's Random surfaces enumerating algebraic curves. So he mentions something about virtual fundamental class....
- 4,466
1
vote
0
answers
207
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understanding some derivation in random XORSAT problem
This question is concerned about the paper "The 3-XORSAT threshold" by O. Dubois, J.Mandler. Here is the link: http://dx.doi.org/10.1016/S1631-073X(02)02563-3
Basically one would like to know when is ...
- 4,466
1
vote
1
answer
807
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simulating chances of success when drawing from a bag of biased coins
I am trying to plot the pdf of flipping heads when drawing from a bag of biased coins. Since I am interested in the % of heads flipped, not the number, I simulate 500K flips and group the results into ...
- 13
9
votes
2
answers
573
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half-plane percolation clusters
Consider critical edge-percolation in the induced subgraph of the square grid with vertex set {$(i,j) \in Z \times Z:\ i+j \geq 0$}, and let $p_n$ be the probability that the cluster containing $(0,0)$...
- 19.7k
2
votes
1
answer
258
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How would one extend the Brier score to an infinite number of forecasts?
Is there a neat way to use something like the Brier score to score an infinite set of forecasts/outcomes?
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8
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1
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6k
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Big picture concerning Ito integral, Stratonovich integral and standard results in probability theory
I am confused and don't get the big picture concerning the connection between
Ito integral
Stratonovich integral
Standard results in probability theory concerning skewed distributions.
Example: Take ...
- 5,935
5
votes
2
answers
641
views
Percolation Model and Complex Probabilities
Let $d>0$ be an integer and consider the first neighbors independent bond percolation model in $\mathbb Z^d$, where each edge is open with probability $p\in[0,1]$.
I would like to know, if can we ...
- 2,044
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votes
2
answers
869
views
How can I calculate the expected ranking of a competitor from the probabilities of each competitor reaching first place?
Say I have several competitors contending over some prize. I know the probabilities that any particular one of them will win the prize. It is assumed that the competitors all want to achieve the ...
4
votes
0
answers
696
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Dynamic programming principle (DPP)
In stochastic control problem, one shall use the measurable selection theorem to prove DPP. It was discussed in discrete time case in [Bertsekas and Shreve 1978]. Is there unified framework in ...
- 1,399
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0
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3k
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Good textbooks on probability and/or stochastic processes, emphasizing simulation
Any recommendations for textbooks on probability and/or stochastic processes that emphasize simulation? I'll be teaching this course in the Fall.
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1
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837
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"Nice" Solution to repeated integral
I have a problem wherein I have defined a function $I_r(t) = \int e^{(2r-1)at} \int e^{(2r-3)at} \cdots \int e^{at} dt\cdots dt$, and $I_r(0) = 0$, for $r = 1,2,3,\ldots$.
I find that $e^{-ar^2t} I_r(...
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3
answers
5k
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Mean minimum distance for K random points on a N-dimensional (hyper-)cube
Given K points in a N-dimensional (hyper-)cube with all edges length 1.
What is the expected minimal distance between 2 points.
I found the 1-dimensional case in this topic: Mean minimum distance for ...
- 371
1
vote
4
answers
2k
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How is a permutation taken as an equivalent of a hash function in MinWise independent permutations?
In the paper on MinWise independent permutations (MinWise independent permutations), the authors say that it is often convenient to consider permutations rather than hash functions (Pg-3).
While I ...
- 113
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votes
1
answer
3k
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Is there a sigma algebra without atoms on a countably infinite set ?
The title says is all.
To motivate the problem, here is a theorem for finite sets.
Theorem: If S is a finite set, then it can be proved that the atoms of any sigma algebra on S form a partition of S....
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