All Questions
Tagged with pr.probability probability-distributions
1,385 questions
23
votes
5
answers
2k
views
Maximizing the expectation of a polynomial function of iid random variables
Let $f \colon \mathbb R^N \to \mathbb R$ be a smooth function. Let $\mu$ be a probability measure on $[0,1]$ and $X_1, \ldots , X_N$ be i.i.d. random variables on $\mathbb R$.
Question 1. What is ...
- 2,479
1
vote
1
answer
1k
views
Monotonicity, Convexity, and Smoothness of the KL-Divergence between Two Brownian Motions with Different Initializers
We consider the two distributions
$$
p_t = p_0 * N(0, tI),\quad q_t = q_0 * N(0, t I),
$$
where $*$ denotes the convolution between two densities, while $p_0$ and $q_0$ have the same mean and ...
- 6,045
0
votes
1
answer
80
views
Expectation of ratio between product of gaussian r.v.'s and generalized gamma r.v
Given
\begin{equation}\label{eq:definition_of_z}
\begin{split}
\textbf{Z} = \left[\begin{array}{cccc}
{z}_{11} & {z}_{12} & \cdots & {z}_{1P} \\
{z}_{21} & {z}_{22} & \cdots & {...
3
votes
2
answers
430
views
Multivariate normal concentration
If $X\sim N(0,\Sigma)$ for some $d$-dimensional normal distribution, then $X = \Sigma^{1/2} Z$ where $Z\sim (0,I)$. How to compute the following quantity?
$$
\operatorname{var} (X^T X)
=
\...
- 593
1
vote
1
answer
239
views
Probability of two Points being divided by an high-Dimensional Hyperplane
I have two points $x_1,x_2 \in \mathbb S^n $ which are distant $d$ from each other, where $d<<1$.
I also have a vector $v$ sampled uniformly at random from $\mathbb S^n$.
What is the ...
- 128k
0
votes
0
answers
268
views
Taylor series expansion of quantile function
Suppose $Y$ and $Z$ two random variables, $\lambda \in \mathbb{R} $.
We note $F^{-1}_{Y}(\alpha)$ the quantile function of the variable $Y$ at the quantile level $\alpha \in (0,1)$.
Do you have any ...
- 4,729
2
votes
1
answer
403
views
Product of independent random variables and tail deconvolution
Suppose $X, Y$ are two independent non-negative random variables. The conditions
(i) $\mathbb{P}(X > t) = \frac{C}{t^p} + o(t^{-p})$
(ii) $\mathbb{P}(Y > t) = o(t^{-q})$ for any $q > ...
CommunityBot
- 1
1
vote
1
answer
137
views
Approximate $\log \mathbb E_P[\exp(th(x)]$ for a function $h$ which is lipschitz and has finite moments of order 1 and 2 w.r.t $P$
Let $P$ be a probability measure on a space $\mathcal X$ and $h: \mathcal X \rightarrow \mathbb R$ is measurable function with finite moments of order 1 and 2. I'm interested in approximating the ...
- 128k
3
votes
4
answers
451
views
Solution of a 2D Recurrence sequence
Can we solve the following recurrence relation:
$$a_{m,n} = 1 + \frac{a_{m,n-1}+a_{m-1,n}}{2}$$
with $a_{0,n}=a_{m,0}=0$? If not, can we get an estimate of the growth of $a_{m,n}?$
I encountered this ...
- 128k
1
vote
1
answer
90
views
Expectation of random variables coincides
Let $Y_1:=(X_i)_{i \in \mathbb Z}$ be a family of random variables that are identically distributed but not necessarily independent.
We can then also define the shifted sequence $Y_2:=(X_{i+1})_{i \...
- 3,993
11
votes
8
answers
2k
views
Semicircle law universality elsewhere
Wigner's semicircle distribution is:
$$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$
Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...
- 188k
2
votes
1
answer
676
views
Distribution of ratio between complex Gaussian and Chi-square R.V.s
What would be the distribution (p.d.f.) of the following ratio?
$$z = \frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$
where $x_{i} \sim \mathcal{CN}(0,a), \forall i$ and $a > 1$. As can ...
1
vote
1
answer
305
views
Existence of a Lyapunov function for a log-concave measure
Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1$$ and $\mu$ denote the measure with density $e^{-f}$ with respect to the Lebesgue measure on $\...
- 167
1
vote
0
answers
52
views
Stochastic Control: Markovian restriction
Consider a stochastic control problem,
$$v^C(0,x) = \mathbb{E} \Big[\int_0^\tau f(X_t,C_t) d t + (T-\tau)|X_\tau|\Big] $$
where $X_t$ is a weak solution to the SDE
$$dX_t = C_t dB_t, \quad X_0 = x \...
- 553
1
vote
1
answer
186
views
Expected norm of linear maps
I want to compute the expected norm of a vector-matrix multiplication. I have a vector $x \in \mathbb{R}^n$ with norm one and a matrix $M \in \mathbb{R}^{n \times n}$, whose entries are iid taken from ...
- 128k
0
votes
0
answers
424
views
Bounding the total variation distance between two measures from a given set
I have a distance on the space of probability measures on $[0,2]$. It is defined as such for two probability measures $\mu_1$ and $\mu_2$ :
$d_p(\mu_1,\mu_2) := \sum_{k=0}^p ( \mathbb{E}[X_1 ^k]- \...
- 31
0
votes
0
answers
57
views
Absolute continuity of probability measures determined by dependence structure
We are on $\mathbb{R}^d$ with Borel $\sigma$-algebra. Let $\mu_1, ..., \mu_d$ be probability measures on $\mathbb{R}$ and $\Pi(\mu_1, \mu_2, ..., \mu_d)$ be the set of probability measures on $\mathbb{...
- 1,095
4
votes
2
answers
122
views
The minimum of the reciprocals of some Poisson random variables
Let $X_1,\dots,X_k$ denote a collection of independent samples of a Poisson random variable whose mean also happens to be equal to $k$. Does the quantity $$k\boldsymbol{E}\min\left\{ \frac{1}{1+X_{1}}...
- 4,361
8
votes
2
answers
891
views
Differentiating an integral that grows like log asymptotically
Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
...
- 128k
3
votes
2
answers
227
views
Example of measure for some algebra over N
$\mathcal F$ is set of events. Can you give an example of some algebra $\mathcal A$ over $\mathbb N$ and a non-zero finitely additive measure $\mu$ on this algebra, which has a countably additive ...
- 4,991
3
votes
2
answers
231
views
Expectation of the exitpoint distance for the symmetric random walk
Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ such that $\nu(\{0\})\neq 1$.
Consider a random walk started at $S_0=0$, denoted $S_n=X_1+\dotsb+X_n$, ...
- 173
0
votes
0
answers
268
views
Does the linear combination of the quantile $\alpha F^{-1}(\tau)+\beta G^{-1}(\tau)$ still a quantile
$F(x)$ and $G(y)$ are distribution functions.
Define the $\tau$th quantile for cdf $F(x)$, $G(y)$ as
$$\xi_\tau\equiv F^{-1}(\tau)=\inf\{x:F(x)\ge \tau\}$$
and
$$\eta_\tau\equiv G^{-1}(\tau)=\inf\{y:...
- 141
-1
votes
1
answer
370
views
What's the probability of two independent events in time domain?
Suppose there are two independent events A and B. The probability that A or ...
- 188k
3
votes
0
answers
187
views
Are there any conditions on the moments that make a measure a probability measure?
For a positive Borel measure $\mu$ on the real line interval $[-1, 1]$, let $\displaystyle{m_n = \int_{-\infty}^\infty x^n d\mu(x)}$, i.e. the $n$th moments of the measure. Are there any conditions ...
- 210
8
votes
1
answer
171
views
On the existence of a particular type of finite measure on $\mathbb N$
Let $\mathbb N$ denote the set of all positive integers. Does there exist a countably additive measure $\mu : \mathcal P(\mathbb N) \to [0,\infty)$ such that $\mu(\mathbb N)<\infty$ and $\mu(\{nk: ...
- 12.8k
5
votes
1
answer
252
views
What is the pdf of Laplace distribution conditioned on a plane? How can I sample from it?
Our goal is to sample from the Laplace distribution conditioned on a linear subspace. Here are the details of this problem.
Let
$$p(x) \propto \exp(-\|x\|_1/\sigma)$$
be the pdf of the Laplace ...
- 128k
1
vote
1
answer
467
views
Sharp tail bounds for the maximum of an iid sample of a random variable supported on $[0, 1]$
Let $X_1,\ldots,X_n$ be an iid sample from a distribution supported on $[0, 1]$.
Question
What are some sharp concentration inequalities (i.e tail bounds) empirical statistic defined by $Z_n := \max(...
- 128k
5
votes
1
answer
107
views
Estimating the size of the remainder in a random partition
Pick a sequence of real numbers $x_i$ as follows. Put $x_0=1$. If $x_i$ is chosen, then pick $x_{i+1}\in[0, x_i]$ according to the uniform distribution. Obviously we have $x_i\rightarrow 0$ with ...
- 23.2k
3
votes
0
answers
156
views
Probability distribution from equidistribution - I
Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_r(a,b)$ to be minimum $\ell_r$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...
- 13.9k
2
votes
2
answers
74
views
Given two probability density functions find a number that satisfies a given equation
I have a problem for which I either need a proof or a counterexample.
We are given two discrete random variables $x_1$ and $x_2$ in $[0, n]$ where $F_1(x)$ is the probability of $x_1\leq x$, and ...
- 128k
3
votes
1
answer
115
views
Probability density from standard domain - I
Pick $x+iy$ at random with respect to hyperbolic measure from $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$. What does the probability distribution function of $\frac1{\sqrt y}$ look like?
- 13.9k
3
votes
2
answers
259
views
What is $\sum_{k=0}^{+\infty}{k⋅p(k;\mu_1,\mu_2)}$, where $p$ is the pmf of Skellam distribution?
The Skellam distribution is the discrete probability distribution of the difference $N_{1}-N_{2}$ of two statistically independent random variables $N_{1}$ and $N_{2}$, each Poisson-distributed with ...
- 111
1
vote
0
answers
193
views
Calculating the expectation of a sum of dependent random variables
Let $(X_i)_{i=1}^m$ be a sequence of i.i.d. Bernoulli random variables such that $\Pr(X_i=1)=p<0.5$ and $\Pr(X_i=0)=1-p$. Let $(Y_i)_{i=1}^m$ be defined as follows: $Y_1=X_1$, and for $2\leq i\leq ...
- 31
2
votes
1
answer
228
views
Maximum of sums of iid $X_i$'s where $X_i$ is the difference of two exponential r.v
Given $X_i = A_i - B_i$ where $A_i\sim \text{ Exp}(\alpha)$ and $B_i \sim \text{ Exp}(\lambda)$. Define $S_k = \sum_{i=1}^k X_i$ with $S_0 = 0$, and
$$M_n = \max_{1\leq k \leq n} S_k.$$
Is it ...
- 485
2
votes
1
answer
280
views
Complicated bound after using Stirling's approximation
I have this inequality $$\frac{1}{a}\exp\bigl\{-\frac{4}{h^2}\bigr\} \geq \frac{1}{f}$$ where $$ a \leq \Bigl(\pi^{d/2}\Gamma(\frac{1}{2}d+1)^{-1} + 1\Bigr) \left(\frac{h^{d+1}}{2} \Gamma \left(\frac{...
- 225
4
votes
2
answers
316
views
The probability density function of the number of coins to first fill one box of $N$
Given $N$ boxes with the same capacity $C$, I toss coins into the boxes uniformly, one by one. When any one of the boxes is full, the sum of the coins in all boxes is denoted $S$. How to compute the ...
- 3,255
3
votes
1
answer
202
views
Is there a coupling that induces a given coupling via a transition kernel?
Let $X,Y$ be two measurable spaces, $\mu,\nu$ two probability measures on $X$, and $\kappa$ a transition kernel from $X$ to $Y$.
Define $\tilde\mu(dy)=\int_X\kappa(dy|x)\mu(dx)$ and $\tilde\nu(dy)=\...
- 576
12
votes
1
answer
617
views
Mode of a sum of Bernoulli random variables
Let $S_n=\tau_1+\cdots+\tau_n$ be a sum of independent Bernoulli random variables such that $\mathbb{P}(\tau_i=1)=p_i$. Is it true that the mode of $S_n$ is either its mean rounded up or rounded down?
- 4,679
3
votes
2
answers
189
views
Is the covariance of squares always bounded from below by two times the covariance?
I came across the following inequality in one of my calculations ($X,Y$ are centered random variables):
$$\operatorname{E}(X^2Y^2)-\operatorname{E}(X^2)\operatorname{E}(Y^2) \geq 2 \operatorname{E}(...
- 128k
2
votes
2
answers
2k
views
Dependence between direction and magnitude of multivariate normal random vector
Suppose that $x\sim N(0, V)$ is $p$ dimensional with $V$ diagonal having elements $v_i^2$. Then
\begin{align}
f_x(x) & \propto \left(\prod_p v_i\right)^{-1} \exp\left(-\frac{1}{2}\sum_p \frac{x^...
-1
votes
1
answer
76
views
Transforming random variables for having good property?
For arbitrary functions $A$ and $B$ and independent random variables $X$ and $Y$, assume that
\begin{align}
\Omega&\triangleq \{(x,y): A(x,y)=1\},\\
\Lambda&\triangleq \{x: B(x)=1\}.
\end{...
2
votes
0
answers
232
views
Random walk and comparing sums of Exponential random variables
Let $\sigma$ be the time a nearest neighbor random walk started at 1 that has probability $p>1/2$ of moving left reaches $0$. Let $\sigma'$ be an independent copy of $\sigma$. Let $(X_k)_1^\infty$ ...
- 457
1
vote
1
answer
118
views
What is the order of the left tail of a mixture of non-central chi-square?
Let $\mu\sim N(0,1)$, $Z\sim N(\mu,1)$. Then $Z$ can be viewed as a mixture of Gaussians. It can also be viewed as a Gaussian but there is a prior for the mean.
Let $X\sim\exp(\lambda)$ where the ...
CommunityBot
- 1
1
vote
1
answer
154
views
Generalization of inverse transform sampling
If X is a random variable over an arbitrary alphabet, is there a (deterministic) function f() such that X = f(U), where U is a uniform random variable over the unit-interval?
- 128k
13
votes
2
answers
669
views
An inequality for expected value of normally distributed variables
Question. Let $X_1,\dots,X_n$ be random variables with normal distribution. Is it true that
$$\mathbb E \prod_{i=1}^nX_i^{2k}\ge\prod_{i=1}^n\mathbb E X_i^{2k}$$for any $k\in\mathbb N$?
(The ...
- 4,535
4
votes
2
answers
415
views
Effect of perturbing the atoms of a measure on the Wasserstein distance
Let $(X,d)$ be a metric space, $x_1,\ldots,x_N\in X$ and $x_1',\ldots,x_N'\in X$ be atoms, and $G=\sum_{i=1}^Np_i\delta_{x_i}$, $G'=\sum_{i=1}^Np_i'\delta_{x_i}$, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...
- 576
6
votes
4
answers
1k
views
Improvement of Chernoff bound in Binomial case
We know from Chernoff bound
$P\bigg(X \leq (\frac{1}{2}-\epsilon)N\bigg)\leq e^{-2\epsilon^2 N}$ where
$X$ follows Binomial($N, \frac{1}{2}$).
If I take $N=1000, \epsilon=0.01$, the upper bound is ...
- 128k
1
vote
2
answers
462
views
lower bound the probability of at least L collisions
Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$.
If we now ask ...
- 128k
4
votes
1
answer
275
views
A metric stronger than total variation
Let $P,Q$ be two distributions on a finite set $X$. Consider the following metric*
$$ d(P,Q) = \frac12\max_{\emptyset\neq A\subseteq X} \|P(\cdot\mid A)-Q(\cdot\mid A)\|_1. $$
Obviously, the total ...
2
votes
1
answer
210
views
Controlling Mean Difference Between Product and Joint Distributions Using Optimal Transportation
Suppose we have nonindependent random variables $X \sim P$ and $Y \sim Q$, where $P$ and $Q$ denote their marginal distributions. We are interested in upper bounding
$$
|\mathbf{E}_{X, Y\sim P \...
- 128k