All Questions
Tagged with pr.probability probability-distributions
1,384 questions
3
votes
0
answers
134
views
Algorithm to calculate moments of uniform distribution on convex polyhedra
There is system of linear inequalities
$$
Ax \leq K,
$$
$$
x\geq a, x\leq b.
$$
$A$ is $(n\times m)$-matrix, where $n\approx 100$ and $m\approx 10000$, $rank(A)=n$.
Suppose that on set of solutions ...
1
vote
1
answer
219
views
connection between the statistical properties of a scalar field and its columns
Consider a scalar field $s:[0,1]^3 \to \mathbb{R}$ and its "column" field
\begin{equation}
c: [0,1]^2 \to \mathbb{R}: (x,y) \mapsto \int_0^1 s(x,y,z) \,\mathrm{d}z.
\end{equation}.
What can be said ...
- 113
0
votes
0
answers
51
views
derivation of a gap related to extreme value theory
I have an expression to evaluate as follow:
$\mathbb{E}\left[\sum_{k=1}^K s_k f(x_k)\Big|s_k=s_k^{\ast} \right]$
where $\{s_k^\ast\}$ can be treated as a ${policy}$ which is defined as follows:
\...
5
votes
2
answers
982
views
Bounds on $\int \log(1+x) g(x) \mathrm{d}x$?
Let $X$ and $Y$ be two continuous real random variables with common support $(0,x_{\max}]$ and with PDF $f_X(x)$ and $f_Y(y)$. Assume that $\Pr [Y\geq\beta \mid X<\beta] \leq k$ and that $\Pr [Y<...
- 482
6
votes
1
answer
385
views
Functional limit theorem under random change of time
FINAL EDIT: There is one main question left: According to the answer, we have choosen $\theta=1$ , where we could choose $0<\theta<\infty$ as we like. His this sufficient, if we regarde the ...
- 61
3
votes
1
answer
827
views
Solving recursion / finding generating function of a probability mass function
I am assessing the probability distribution on a running time of some algorithm that we've developed. I am looking for a family of probability mass functions $f_n$ with the following recurrence:
$$
f_{...
- 396
2
votes
2
answers
182
views
Difference between maxima of random variables
Given four independent, identically distributed Gaussian random variables with zero mean and unit variance $x_1$, $x_2$, $y_1$, $y_2$, consider
\begin{equation}
u \equiv \max(x_1+C\, y_1, x_2+C \, ...
- 151
3
votes
1
answer
187
views
Moment matching on the standard simplex
Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where
$$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j \...
- 1,127
3
votes
1
answer
269
views
Learn a distribution from distributions on samples
There's many good ways to learn a distribution $p_X$ of an r.v. $X$ over $k$ symbols given many i.i.d. samples $X_1,\ldots, X_n$. The simplest is to use the sample relative frequencies $\hat{f}_X$ as ...
- 131
1
vote
0
answers
102
views
Stochastic Ordering of Negative Binomial-like Distributions
Please forgive me if this is not precise enough to post here. Simply ask me to remove it if it is not suitable. I am new here.
I am bounding the running time of an algorithm as a random variable $X$ ...
- 11
3
votes
0
answers
698
views
How does Jensen Shannon divergence and KL divergence correlate?
I am wondering if there is way to derive the correlation between Jensen Shannon divergence and KL divergence for two distributions: P and Q, in order to show that if JSD(P,Q) decreases, KLD(P,Q) ...
- 31
9
votes
1
answer
556
views
Berry-Esseen bound for martingale sequence with varying and dependent variances
Let $(X_{1},\ldots,X_{k},\ldots)$ be a martingale difference sequence, i.e.
$$
E[X_{k}|\mathcal{F}_{k-1}] = 0
$$
where $\mathcal{F}_{k-1}$ is the $\sigma$-algebra filtration at $k-1$.
Let $\sigma_{...
- 719
6
votes
1
answer
3k
views
expected value of multiplication of matrices
I start with background and then ask my question, background is a brief description of wishart distribution.
Background
The Wishart distribution with $\nu$ degrees of freedom and positive definite $...
- 61
6
votes
0
answers
277
views
universality for large deviations?
This is a question about universality in probability theory, with combinatorics in mind.
Consider a sequence of polynomials $P_n$ in one variable, with positive coefficients. Combinatorics is a large ...
- 3,587
3
votes
0
answers
276
views
Processes with the same finite dimensional distributions as the solutions to SDEs
Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...
- 131
5
votes
1
answer
1k
views
Supremum of a martingale
Let $(X_n)$ be a martingale. What can be said about the distribution of its maximum over a window of fixed length:
$$M_n = \max_{n-10 \leq k \leq n} X_k$$ or about the "range" over a window:
$$R_n = \...
- 587
7
votes
2
answers
605
views
Gaussian and the convex hull of moment curves
Let $c_1,\dots, c_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c_1,\dots, c_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a ...
- 1,127
8
votes
2
answers
2k
views
Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian
This is a question related to the statistical model behind independent component analysis (ICA).
We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...
- 1,127
1
vote
1
answer
148
views
Distribution of maximum unique number of several random numbers
Suppose discrete random variables $\{X_1, X_2, ..., X_n\}$ are i.i.d. described by the probability function:
$f(x) \equiv \text{Pr}(X_i = x)$,
and $X_i \in \{1,2,3, ..., m\}$.
Let $Y$ be the ...
- 399
4
votes
1
answer
220
views
Question about the weak convergence of probability
Let $\mu$ be a probability measure on $\mathbb R$ and set
$$c(K):=\int_{\mathbb R}(x-K)^+d\mu(x).$$
Assume that one has a sequence of probability measures $(\mu_n)_{n\ge 1}$ s.t.
$$\int_{\mathbb R}\...
- 1,835
1
vote
1
answer
166
views
Question abouth Skorokhod representation of random variables (II)
This is a continuation of
Question abouth Skorokhod representation of random variables
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that
$$\int_{\mathbb R}|x|^pd\mu(x),~ \...
- 1,835
5
votes
1
answer
356
views
Question abouth Prokhorov metric
Let $X$ and $Y$ be two random variables with first order moments, i.e. $E[|X|]$, $E[|Y|]<+\infty$. Assume further that
$$E\left[|X-Y|\right]<\varepsilon.$$
Set $Law(X)=\mu$ and $Law(Y)=\nu$, ...
- 1,835
2
votes
2
answers
1k
views
divisibility of uniform distribution [closed]
Let $X$ and $Y$ be independent and identically distributed random variables.
Can $X+Y$ be a uniform distribution?
(Please prove.)
In other words, is a uniform distribution divisible?
The meaning of "...
3
votes
1
answer
304
views
Question abouth Skorokhod representation of random variables
It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e.
$$\rho(\mu,\nu)<\varepsilon,$$
then there exist two random ...
- 1,835
2
votes
0
answers
619
views
Laplace transform of a integral function of CIR/CEV process
The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...
- 323
5
votes
2
answers
559
views
A measure of how "spread out" a probability measure is
Consider a random variable $X$ whose variance is large. As a contrast to Markov's or Chebyshev's inequality, both of which measure the concentration of a probability distribution, is there a measure ...
- 527
5
votes
1
answer
297
views
Random walk with continuously distributed steps on [-1,1]
A simple random walk $S_n = X_1 +\cdots +X_n$, where $P(X_i = 1) = p \not = 0.5$ and $P(X_i=-1)= q \triangleq 1-p$, admits the following probability
$$P(S_n \textrm{ reaches } a \textrm{ before} -b) =...
- 108
1
vote
1
answer
290
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
vote
0
answers
82
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 ...
- 139
2
votes
1
answer
443
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 $...
user83947
2
votes
1
answer
886
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 X_i}{\...
- 143
2
votes
0
answers
208
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 ...
- 482
1
vote
0
answers
80
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 $\mathsf{var}(...
- 1,109
-1
votes
1
answer
213
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 ...
- 482
4
votes
1
answer
229
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 ...
- 161
2
votes
1
answer
99
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 ...
- 457
2
votes
1
answer
461
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
$$\mathbb E[f(X_t)] = \mathbb E[f(\hat ...
- 1,399
11
votes
1
answer
283
views
Probability distribution derived from gamma function - does it have a name?
Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$.
Now, let's fix $\sigma$ and let t vary. Then consider the following expression:
$$|\Gamma(\sigma+it)|^2$$
For any choice of $\...
- 4,897
1
vote
1
answer
606
views
moment sequence which does not define a random variable vs convergence in distribution
I am encountering the following problem concerning existence of a limiting random variable (in distribution): assume a sequence of positive random variables $\{X_n\}_{n\geq 0}$ from which we know ...
- 1,561
1
vote
1
answer
139
views
Exponentially Bounded Sequence of Moments defining Distribution?
I have an exponentially bounded sequence $m_n = \lambda^n + c_n$ (i.e. the $c_n$ are quadratic in $n$) and would like to know if this sequence of moments defines a distribution. I considered applying ...
user82426
2
votes
2
answers
435
views
Convergence in distribution to a Poisson
We have encountered the following problem that we think that should be true. Let $\{X_n\}_{n\geq 0}$ a sequence of random variables which we know that $\mathbb{E}[X_n]$ tends to infinity.
The ...
- 1,561
6
votes
2
answers
378
views
Slight variation on law of the iterated logarithm
Let$$M_t = \max\{B_s : 0 \le s \le t\},\text{ }m_t = \min\{B_s : 0 \le s \le t\},$$where $B_t$ is a standard Brownian motion. My question is, does there exist $r$ such that with probability one,$$\...
user80013
3
votes
1
answer
380
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.
$$P(X_1=n_1,\ldots,X_k=n_k) = {n\choose n_1,n_2,\ldots,n_k} p_1^{n_1}...
- 1,269
2
votes
1
answer
571
views
Extension of Dynkin's formula, conclude that process is a martingale
This question was asked here, but it did not get enough attention, so I'm crossposting it to MO.
Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial ...
user61522
0
votes
0
answers
260
views
Concluding that the Poisson kernel is indeed the Cauchy distribution?
See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\...
- 13
1
vote
1
answer
237
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$$E^{(x, y)}\...
- 13
0
votes
1
answer
186
views
Poisson kernel, $E^{(x, y)}\text{exp}\{i\theta X_t - \theta Y_t\} = e^{i\theta x - \theta y}$
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. How do I see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\text{exp}\{...
user76167
6
votes
2
answers
2k
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 ...
- 61
2
votes
1
answer
157
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?
user75246
-1
votes
1
answer
519
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?
