All Questions
Tagged with pr.probability probability-distributions
1,384 questions
1
vote
0
answers
190
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Nontransitive dice: the least number of faces?
Here is an introduction to nontransitive dice. The question is: given $n$-player with a $m$-sided dice each one, the what is the minimum of $m$ for a fixed $n$ to produce nontransitivity?
Here is ...
CommunityBot
- 1
2
votes
0
answers
341
views
Marginalizing multivariate normal over defined interval
Hello everyone,
I am trying to obtain an analytic expression for the following Gaussian integral
$$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} \;...
- 101
1
vote
1
answer
478
views
Distance between the product of marginal distributions and the joint distribution
Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose:
\begin{align}
P1(A,B,C) &= P(A) P(B) P(C) \\
P2(A,B,C) &= P(A,B) P(C) \\
P3(A,B,C) &= P(...
- 49
5
votes
1
answer
765
views
Measure concentration for weakly dependent random variables
For an application quite alien to probability theory, I'd like to have a kind of measure concentration estimate, in the following spirit. Suppose that to every $1\le i,j\le n$ there corresponds a zero-...
CommunityBot
- 1
7
votes
1
answer
3k
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What is characteristic function of maximum of i.i.d. random variables?
Is is possible to get characteristic function of maximum of i.i.d. random variable sequence? Such as $X_1, X_2$ are two i.i.d random variables, then what is characteristic function of $X=\max(X_1,X_2)$...
- 3,968
0
votes
2
answers
174
views
Joint distribution with specified marginals
Suppose we are given a probability distribution over a finite discrete product space $p(x,y)$ with marginals $p(x), p(y) > 0$ for each $x,y$ respectively. We are given two more marginal ...
- 37.7k
1
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0
answers
1k
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Expected value of the log of the factorial of a poisson distribution
I found the expression for the expected value of the falling factorial of a Poisson distribution ($\lambda^n$) from - http://en.wikipedia.org/wiki/Factorial_moment. Is there a similar expression for ...
- 6,210
4
votes
2
answers
1k
views
Probability distribution over cluster size in Erdős–Rényi random graph.
My question is about the probability distribution over the possible size of the containing cluster of a randomly chosen node in an Erdős–Rényi random graph.
Let G(n,p) be an Erdős–Rényi random graph (...
- 248
4
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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 ...
CommunityBot
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-2
votes
1
answer
292
views
Probability distribution needed [closed]
Let me clarify my needs. The PDF must comply to:
1. The mean is always in the shorter tail
2. Should have an inverse function
3. Be defined in the interval [0, 1]
4. Should have a shape parameter that ...
CommunityBot
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1
vote
1
answer
228
views
Is anything known about Large Deviation Principle for non additive functionals on Markov chains?
Let $\Sigma$ be a finite set of cardinality $|\Sigma |$ and
$$\Pi = \{ \pi(i,j)\}_{i,j = 1}^{|\Sigma|}$$
a stochastic matrix (ie a matrix whose elements are non negative and such that
each row sum ...
- 7,514
7
votes
1
answer
12k
views
inner product of two gaussian random vectors?
Suppose that $x, y\sim N(0,I_n)$ are independent. Consider the inner product $\langle x, y\rangle$. Intuitively, $y$ behaves like a random vector of length $\sqrt n$, so $\langle x, y\rangle$ is close ...
- 802
1
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0
answers
442
views
Joint distribution from multiple marginals
Consider an experiment consisting of a repeated trial with two random Bernoulli (=binary) variables, A and B. Each trial consists of multiple outcomes for both A and B. Each trial has the same number ...
- 113
2
votes
2
answers
869
views
Computing hypergeometric function of matrix argument
In the context of the Bingham probability distribution the ${ }_1F_1$ hypergeometric function of matrix argument naturally arises as a normalization constant of the probability distribution function. ...
CommunityBot
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4
votes
1
answer
700
views
Bounding statistical distance by matching moments
Suppose we have distributions $p(x)$ and $q(x)$ both supported on integers in $[-n, +n]$. We want $p$ and $q$ to have statistical (total variational) distance of at most $\epsilon$.
Is there a ...
CommunityBot
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3
votes
0
answers
104
views
Minimizing/Maximizing the tail of the convex combinations of Chi Squared i.i.d random variables
Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors:
\begin{eqnarray*}
\bar{a} &=& (\frac{1}{...
- 51
2
votes
0
answers
198
views
Have you seen this one parameter family of distributions before?
This is a one parameter family of distributions. Choose some parameter $\lambda > 0$ and define the measure $\nu_\lambda$ which is absolutly continuous with respect to the Lebsegue measure with the ...
- 21
0
votes
1
answer
229
views
Weak convergence in measure for negligible sets.
Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
- 17k
2
votes
0
answers
141
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question about circular law
Hi,
I have a question about the circular law.
Consider $A_n=[x_{ij}]$ a sequence of random matrices where $x_{ij}$ are iid with mean $0$ and variance $1$. Consider $\lambda_{n,1},\dots,\lambda_{n,n}$ ...
- 29
3
votes
1
answer
354
views
Central Limit Theorem for additive function of permutations of sequences
Fix the finite sets $\mathcal{X}$ and $\mathcal{Y}$, and probability mass functions $P_X(x)$ and $P_Y(y)$ on these sets. Assume each value of $P_X(x)$ and $P_Y(y)$ is rational.
For each $n$ such ...
- 325
0
votes
2
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298
views
Are all variables in a set of random variables independent if all pairs are independent?
If I have a sequence of random variables $X_1, X_2, \ldots, X_n$ (possibly infinite) such that all pairwise cdf's are factorized:
$$F(X_i, X_j) = F_i(X_i) F_j(X_j)$$
for all pairs $(X_i, X_j)$, does ...
- 37.7k
2
votes
1
answer
2k
views
Is an L_1 bounded sequence of random variables with uniformly converging CDFs uniformly integrable?
Changing my question in light of Dan's answer. Thanks, Dan.
Consider a sequence of real random variables $X_i$ bounded in $L_1$, that is $\mathbb E\left|X_i\right|\leq M$ for all $i$. Suppose that ...
- 23
1
vote
2
answers
4k
views
minimum of different independent Poisson random variables
Let $X_1,\ldots,X_N$ be independent Poisson distributed random variables with unequal parameters $\lambda_1,\ldots,\lambda_N$.
Is there any closed form expression or at least a good approximation for ...
CommunityBot
- 1
3
votes
3
answers
379
views
Support of an infinitely divisible measure.
Hello,
if $G$ is a compact Lie group. Let $\mu$ be an infinitely divisible measure on $G$, such that $e$, the neutral element of $G$, is in the support of $\mu$. Is that true that the support of $\...
- 2,201
0
votes
1
answer
320
views
Simple markov chain problem
I know this is an easy problem, but I can't figure it out.
A particle takes discrete steps $σ_1,σ_2,σ_3,…,σ_n$ which take on values +1 or −1. However, $P(σ_i=+1)=p$ and $P(σ_i=−1)$ will be $1-p$.
...
CommunityBot
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0
votes
1
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121
views
Multinomial -- how many trials in order to see all the values with prob 1-\alpha
Let suppose that I have a box with $k$ different balls, each one with a different color.
At each time I have to extract a ball and observe the color. Then I put the ball back in the box.
How many ...
1
vote
1
answer
140
views
Equivalence between choosing a subspace and choosing its orthogonal
Hi,
We consider subspaces of $\mathbb{R}^N$.
Suppose that we have a property called $\mbox{Prop}$ that apply to subspaces of $\mathbb{R}^N$. That is to say a function from the set of subspaces of $\...
- 60.5k
6
votes
1
answer
333
views
Trasportation metric (AKA Earth-Mover's, Wasserstein, etc.) as "natural" / "induced"?
Context: Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$_1$ distance), I need to define the natural or canonical metric on the set of all probability distributions ...
CommunityBot
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0
votes
1
answer
333
views
Limit of the stochastic process at time 0
This is not a homework question so please be kind not to remove it right away. I am working on some research but have to justify the following argument: Assume $S_t$ is a continuous stochastic process,...
- 165
0
votes
0
answers
2k
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Derivative of the most probable value (of a gaussian variable) VS most probable value of the derivative
Let $x$ be a random variable with gaussian probability distribution $P(x)$. We assume that $x$ depends parametrically on a parameter $t$ so that :
$P(x(t))=\frac{1}{\sqrt{2\pi\sigma^2(t)}}\exp(-\frac{(...
- 1
4
votes
2
answers
315
views
Sampling from a recursively defined distribution
I'd like to know if there are techniques for sampling from a recursively defined probability distribution, assuming that solving the recursion for a formula for the distribution is too difficult.
As ...
- 131
2
votes
1
answer
469
views
If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
- 586
1
vote
1
answer
368
views
Product of probability densities of the form x^{-t} exp (-ax)
I have two probability distributions $p(x) = N_1 x^{-\tau} \exp(-\frac{x}{x_0})$ and $p(y) = N_2 y^{-\kappa} \exp(-\frac{y}{y_0})$. $N_1$ and $N_2$ are just normalization constants and $x>0$, $y>...
- 54.2k
0
votes
0
answers
104
views
Proving that a property holds for random sequences with given marginal distribution by rearrangement
I am currently investigating the property of random sequences with a special marginal distribution function $F(x)$. Given any random sequence $X_1, X_2, \cdots, X_n$, supposing their joint ...
- 21
4
votes
2
answers
8k
views
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 ...
1
vote
2
answers
147
views
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 ...
- 586
1
vote
1
answer
902
views
Product of densities of a wrapped normal distribution
The density of a wrapped normal distribution is given by
$$\frac{1}{\sigma \sqrt{2\pi} }\sum _{k=-\infty }^{\infty }\text{Exp}\left[\frac{-(\theta -\mu -2\pi k)^2}{2\sigma^2}\right]$$
Considering two ...
CommunityBot
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1
vote
0
answers
501
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Distribution of random vectors
Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$).
A vector $u\in ...
- 143
3
votes
1
answer
520
views
Results regarding $E[\min X,Y]$. when $X$ and $Y$ are independent, of given distributions.
Working on fairly unrelated stuff, I needed to prove the following, fairly easy results, and I wonder if anyone can provide references to the literature. Not being a probabilist I wouldn't know where ...
- 9,756
0
votes
0
answers
111
views
Stationarity of an Integral Process
Let $f$ be a continous deterministic function defined on $\left[0,c\right]$ and $(B_{t}^{H})_{t\geq 0}$ be a fBM with $H\in \left(0,1\right)$. We define a Process $\left(X_{t}\right)_{t\geq 0}$ with
$$...
- 28k
1
vote
0
answers
223
views
Why this two model have same probability distribution?
(1)
Consider the following method of generating a random tree with $n$ nodes.
First expand the root node into two branches.
Then expand one of the two terminal nodes at random.
At time $k$, ...
- 351
3
votes
2
answers
351
views
Continuity of hitting distributions
Hi everybody
Let $U$ be the domain (as shown in the picture) and $\bar{U}$ its closure, further more set $\partial_r U$ to be the reflecting boundary and $\partial_a U$ the absorbing one. The process ...
- 279
15
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2
answers
10k
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Convergence of moments implies convergence to normal distribution
I have a sequence $\{X_n\}$ of random variables supported on the real line, as well as a normally distributed random variable $X$ (whose mean and variance are known but irrelevant). I know that the ...
- 5,005
3
votes
1
answer
376
views
The degrees in a random subgraph
Fix some positive integers $N$ and $d_k$, $k=1,2,\dots$ with $N=\sum_{k=1}^\infty d_k$.
Suppose you have a graph $G$ taken randomly uniformly among the set of all (unoriented) graphs with $N$ ...
- 28k
2
votes
1
answer
272
views
Derivative of the CDF of a family of random variables
Suppose I have a r.v. $Z = X + \alpha Y$ and that $F_Z$ is the probability distribution function of $Z$. If we think of the probability $p = F_Z(q) = \mathbb{P}(X+\alpha Y < q)$ as a function $p = ...
- 54.2k
6
votes
2
answers
4k
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tight bounds on probability of sum of laplace random variables.
Are there tight upper and lower bounds on the density of the sum of $n$ i.i.d laplace random variables that depend on $n$ and the individual laplacian densities?
- 301
6
votes
2
answers
410
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If Mean Residual Lifetime is approximately constant, Residual Lifetime is Approximately Exponential in a Strong Sense
Suppose the "mean residual lifetime," $\mathbb{E}[X-x|X≥x]$ is approximately constant for large $x$. Then, I believe that the conditional tail distribution is approximately exponential, in the sense ...
CommunityBot
- 1
8
votes
3
answers
789
views
A Variance-Tail Description for Continuous Probability Distributions
Start with a continuous probability distribution given by a density function f(x). Let X be a real random variable whose distribution is given by the probability distribution.
I would like to ask ...
- 401
0
votes
1
answer
329
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Is it known that every PDF continuous in all $R^n$ has a maximum? [closed]
I'm working with maximum a posteriori estimation and managed to show that every probability density function that is continuous in all $R^n$ always has at least one global maximum. I've search around ...
- 31.5k
2
votes
1
answer
521
views
Limit of a rescaled random sum of i.i.d. random variables
Consider a sequence of i.i.d. random variables $(X_i)_{i \in \mathbb N}$ and let $S_n=X_1+\dots+X_n$
For every $\alpha \in ]0,+\infty[$, let $N(\alpha)$ be a discrete random variable on $\mathbb N$, ...
- 418