In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

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109 views

Question abouth Prokhorov metric

Let $X$ and $Y$ be two random variables with first order moments, i.e. $E[|X|]$, $E[|Y|]<+\infty$. Assume further that $$E\left[|X-Y|\right]<\varepsilon.$$ Set $Law(X)=\mu$ and $Law(Y)=\nu$, ...
0
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2answers
121 views

divisibility of uniform distribution [closed]

Let $X$ and $Y$ be independent and identically distributed random variables. Can $X+Y$ be a uniform distribution? (Please prove.) In other words, is a uniform distribution divisible? The meaning of ...
3
votes
1answer
155 views

Question abouth Skorokhod representation of random variables

It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e. $$\rho(\mu,\nu)<\varepsilon,$$ then there exist two random ...
0
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0answers
67 views

$\sum_{n\in\mathbb{Z}^2}d\mu(x-2\pi n)=0\Rightarrow$ the summands are pairwise mutually singular

Let $\mu$ a finite measure supported by $\Gamma$ (smooth curve in $\mathbb{R}^2$) and absolutely continuos with respect to the arc length measure on $\Gamma$. Please why if ...
1
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0answers
25 views

steady state of a continuous-time birth-death process

we consider a continuous-time birth-death process $\{X(t),t\geq 0\}$ with discrete state space taking non0negative integer values $\{0,1,2,3,...\}$. The transition rates of the process $\{X(t)\}$ are ...
1
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0answers
22 views

distribution on arriving distinations within certain time [closed]

a traveller is travelling on a map. arriving every vertex of the map, the traveller could choose to go to next vertex according to a constant probability. The probabilities are represented in a matrix ...
2
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0answers
83 views

Laplace transform of a integral function of CIR/CEV process

The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...
6
votes
2answers
344 views

A measure of how “spread out” a probability measure is

Consider a random variable $X$ whose variance is large. As a contrast to Markov's or Chebyshev's inequality, both of which measure the concentration of a probability distribution, is there a measure ...
4
votes
1answer
165 views

Random walk with continuously distributed steps on [-1,1]

A simple random walk $S_n = X_1 +\cdots +X_n$, where $P(X_i = 1) = p \not = 0.5$ and $P(X_i=-1)= q \triangleq 1-p$, admits the following probability $$P(S_n \textrm{ reaches } a \textrm{ before} -b) ...
0
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0answers
35 views

Stochastic dominance for subsets

The subsets of a set $N=\{1,2,\ldots,n\}$ form a lattice, with larger sets being higher up, and a subset $B$ connected to another subset $A=B\cup\{x\}$ (for any $x\not\in B$) higher up by a "pipe". ...
2
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1answer
182 views

Topologies for which the ensemble of probability measures is complete

I have been struggling quite a bit with reconciling my intuitive understanding of probability distributions with the weird properties that almost all topologies on probability distributions possess. ...
1
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0answers
37 views

Marginal of mean from product of student-t and gamma

Let's say we have a distribution with PDF described by the product of Gamma and Student-t distributions. This is equivalent to a generative model, in which precision is first drawn from Gamma, and the ...
2
votes
1answer
91 views

Variance of the normal CDF [closed]

Several threads (e.g. Integration of the product of pdf & cdf of normal distribution ) have shown that $E[\Phi(x)]=\Phi(\mu/\sqrt{\sigma^2+1})$ when $x\sim N(\mu,\sigma^2)$. I'd like to compute ...
0
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0answers
48 views

Majorization in distributions of subsets

The subsets of a set $N=\{1,2,\ldots,n\}$ form a lattice, with larger sets being higher up, and a subset $B$ connected to another subset $A=B\cup\{x\}$ (for any $x\not\in B$) higher up by a "pipe". ...
1
vote
1answer
103 views

Literature question on the convergence rate of the empirical distribution

Assume that given $n$ i.i.d samples $(X_1, X_2, ..., X_n)$ drawn from $p_X$, an unknown probability mass function defined over a finite alphabet $\mathcal{X}$, one wants to estimate $p_X(x)$ for each ...
0
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0answers
33 views

Probability distribution/ Statistics of noise Eigenvectors

What is the distribution of noise Eigenvectors of the sampled covariance matrix? For the signal Eigenvectors the expressions are well defined which approach infinity in the noise Eigenvectors case; ...
2
votes
1answer
139 views

Asymptotic behavior of a ratio of sums of iid random variables

Let $X_i$ and $Y_i$ be distributed identically to $X$ and $Y$, respectively. Assume both $X$ and $Y$ take strictly positive values. Consider the random variable $R_n \doteq \frac{\sum_{i=1}^n ...
2
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0answers
92 views

On the Bhattacharyya distance

Let $X$ and $Y$ be two continuous random variables with support $\mathbb{R}^{+}$ and with PDF $f(x)$ and $g(y)$. If the Bhattacharyya distance of $f$ and $g$ is less than $\epsilon$, then is there any ...
1
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0answers
54 views

A variance-preserving Boolean function [closed]

Let a random variable $X$ be given with $P_X$ supported over $\mathcal{X}$. What are the necessary conditions for the existence of a boolean function $f:\mathcal{X}\to \{0,1\}$ such that ...
-1
votes
1answer
154 views

Regarding a new divergence function of two probability distributions

Let $X$ and $Y$ be two continuous random variables with common support and with PDF $f(x)$ and $g(y)$. For any $0 \leq \alpha \leq 1$, and any constant $\beta$ within the support of $X$ and $Y$ such ...
0
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0answers
79 views

Probability distribution associated with total divisors of an integer

Is there a generalization to https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem which gives distribution function for $$\omega(n)=\big|\{d\in\mathsf{prime}:d|n\}\big|$$ where ...
4
votes
1answer
186 views

How are the real-space RG transformations defined?

I'm reading Shang-keng Ma's book Modern theory of critical phenomena, and I'm a bit confused as to how the real-space RG transformations are defined. Ma basically says that these transformations are ...
0
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0answers
29 views

Bayesian model to estimate the parameter of a Bernoulli law

Suppose we have iid boolean variables $X_1,...,X_T = X_{1:T}$ and the associated deterministic parameters $k_1,...,k_T=k_{1:T}$ and $c_1,...,c_T=c_{1:T}$, where for each $t \in \mathbb{N}$, $k_{t} \in ...
0
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1answer
328 views

What is the derivative of this integral?

I have asked this question here http://math.stackexchange.com/questions/1536018/how-to-find-derivative-of-this-intergral but still has no response. Might I ask it here ? Let $\alpha(t)\in\{0,1\}: ...
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0answers
59 views

Monte Carlo Simulation - efficient simulation of tail outcomes [closed]

When running Monte Carlo type simulations in situations where you're only interested in tail outcomes, do you know of a way to only simulate those outcomes, so that you can come up with more reliable ...
2
votes
1answer
40 views

Conditioned binomial dominates unconditioned with different parameter

Let $X \sim \text{Bin}(n,p)$ and $Y \sim \text{Bin}(n-1,p)$ with $n >1, p \geq 1/2$ and $X,Y$ are independent. I'd like to show $$(X\mid X \geq 1) \succeq_{sd} 1 + Y.$$ Here $(X \mid \cdot)$ is the ...
0
votes
1answer
131 views

Is it safe to work on a Cadlag modification of a Feller process?

Let $f$ be a continuous bounded function. $X$ is a Feller process, and $\hat X$ is its Cadlag modification. By the definition of the modification, one can write $$\mathbb E[f(X_t)] = \mathbb E[f(\hat ...
3
votes
1answer
141 views

density of distance between points in unit circles

Let $a$ and $b$ be two points in the plane. Let's choose a point $c$ uniformly from the circle of radius $r$ with $a$ as center and choose a point $d$ uniformly from the circle of radius $r$ with $b$ ...
10
votes
1answer
164 views

Probability distribution derived from gamma function - does it have a name?

Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$. Now, let's fix $\sigma$ and let t vary. Then consider the following expression: $$|\Gamma(\sigma+it)|^2$$ For any choice of ...
1
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1answer
134 views

moment sequence which does not define a random variable vs convergence in distribution

I am encountering the following problem concerning existence of a limiting random variable (in distribution): assume a sequence of positive random variables $\{X_n\}_{n\geq 0}$ from which we know ...
1
vote
1answer
83 views

Exponentially Bounded Sequence of Moments defining Distribution?

I have an exponentially bounded sequence $m_n = \lambda^n + c_n$ (i.e. the $c_n$ are quadratic in $n$) and would like to know if this sequence of moments defines a distribution. I considered applying ...
0
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0answers
41 views

Convergence of generalized inverses

During the reading about Fisher–Tippett–Gnedenko theorem (it can be found easily on wiki - I don't have enough reputation to post more links), I've got stuck, trying to understand more deeply one of ...
2
votes
2answers
121 views

Convergence in distribution to a Poisson

We have encountered the following problem that we think that should be true. Let $\{X_n\}_{n\geq 0}$ a sequence of random variables which we know that $\mathbb{E}[X_n]$ tends to infinity. The ...
10
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2answers
223 views

Slight variation on law of the iterated logarithm

Let$$M_t = \max\{B_s : 0 \le s \le t\},\text{ }m_t = \min\{B_s : 0 \le s \le t\},$$where $B_t$ is a standard Brownian motion. My question is, does there exist $r$ such that with probability ...
3
votes
1answer
83 views

Uniform convergence of 2-norm of a multinomial vector

Let $(X_1,X_2,\ldots,X_k)$ be distributed according to a multinomial distribution with parameters $(n;p_1,p_2,\ldots, p_k),$ i.e. $$P(X_1=n_1,\ldots,X_k=n_k) = {n\choose n_1,n_2,\ldots,n_k} ...
4
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1answer
75 views

What class of probability distributions do probabilistic turing machines induce? [closed]

What class of probability distributions is induced by the class of probabilistic turing machines? Is there a precise characterization?
0
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49 views

methods to analyze martingale conditioned on return in the future

Consider a martingale $S_t$ on $\mathbb{Z}$ starting from 0. Assume that for any $t$, $Var[s_t\, | \, \mathcal{F}_{t-1}] < V$, where $V$ is some positive constant. Fix an $n$ and for $t \leq n$, ...
10
votes
1answer
294 views

Extension of Dynkin's formula, conclude that process is a martingale

This question was asked here, but it did not get enough attention, so I'm crossposting it to MO. Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial ...
4
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0answers
77 views

Concluding that the Poisson kernel is indeed the Cauchy distribution?

See here. Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, ...
3
votes
1answer
138 views

Poisson kernel, expectation, an absolute value comes in

See here. Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, ...
2
votes
1answer
119 views

Poisson kernel, $E^{(x, y)}\text{exp}\{i\theta X_t - \theta Y_t\} = e^{i\theta x - \theta y}$

Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. How do I see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, ...
7
votes
2answers
262 views

Distribution of $\max_{n \ge 0} S_n$, random walk

Say we have a random walk that is a nearest neighbor random walk on the integers where at each step the probability of moving one step to the right is $p$ and the probability of moving one step to the ...
3
votes
1answer
115 views

Is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane almost surely?

Let $d = 2$. With probability $1$, is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane?
1
vote
1answer
115 views

Formula for maximum of two Gumbel distributions?

I have parameters of two Gumbel distributions ($\mu_1, \beta_1)$ and $(\mu_2, \beta_2)$. Since max of 2 Gumbels is a Gumbel, I'd like to compute $\mu_m, \beta_m$, so that: $Gumbel(\mu_m,\beta_m)$ = ...
2
votes
1answer
126 views

Poisson kernel is the Cauchy distribution, reference?

Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?
1
vote
0answers
28 views

Importance sampling for bernoulli-sequence, favouring long sequences of ones

Assume we have a sequence of i.i.d. bernoulli-distributed random variables of length $n$. I'm interested in doing rare event simulation and my event depends, among other random factors, on the ...
7
votes
1answer
197 views

Brownian motion, crossing intervals, possible usage of second moment method?

This is a followup to my question here. Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le ...
11
votes
4answers
424 views

Number of intervals needed to cross, Brownian motion

Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le {j\over{2^n}}\right\},$$and let$$K_n = \sum_{j = 2^n + 1}^{2^{2n}} ...
0
votes
0answers
36 views

Order statistics of the diagonal terms of an inverse Wishart matrix

I have a question about the inverse Wishart matrix. In my understanding, consider $\mathbf H$ is a $n\times n$ matrix with each elements are complex Gaussian with zero mean unit variance. Then ...
8
votes
1answer
216 views

If $X∼F_1$, $Y∼F_2$, under what conditions on $F_1$, $F_2$ can we construct $Y=E(X\mid\mathscr{G})$ for some $\mathscr{G}$?

Suppose that we have distributions $F_1 $ and $F_2$. Under what conditions on $F_1,F_2$ is it possible to construct random variables $X\sim F_1,Y\sim F_2$ such that $Y=E(X|\mathscr{G})$, that is, $Y$ ...