# Questions tagged [probability-distributions]

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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### Dependence of the integrals of pseudo-Gaussian densities' derivatives on the variance

Let
\begin{eqnarray}
p(t,x,s,y)&:=&\phi(A(t,s,y),y-x),\quad \mbox{with } A(t,s,y)~:=~\int_t^s \frac{\sigma(u,y)^2}{\big(1+\alpha(u)\big)^2} d u \\
q(t,x,s,y)&:=&\phi(B(t,s,y),y-x),\...

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### On an integral of Gaussian CDFs

Let $c>0$ and $T>0$ be fixed. Denote by $F$ the Gaussian CDF, i.e. $F:\mathbb R\to\mathbb R$ is defined by
$$F(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-z^2/2}dz.$$
For every $a\in [0,1)$, ...

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2
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### Cumulants of a sequence of variables with zero mean and variance

Can one prove for a sequence of positive random variable $X_{n}$ such that $\lim_{n\to \infty}E[x_{n}] = 0$ and $\lim_{n\to \infty}E[x_{n}x_{n}]= 0$ all the cumulants go to zero once $n\to \infty$ ?

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### Has the mixture of forward and backward finite difference existed?

Given a function $ f(x) $, there are forward and backward finite differencs, whose definitions are given in the following. By forward one, we mean $ \Delta f(x) = f(x+d)- f(x) $, $(d>0)$; and by ...

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83
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### How to calculate the dual spaces of the following spaces?

Let us consider the spaces $C_\infty(E)$, $C_c(E)$, $C_b(E)$, where $E$ is locally compact, $C_\infty(E）$ is all continuous functions vanishing at the ends of $E$, $C_c(E)$ is all the continuous ...

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### Does the the equivalence of Total variation distance formulas assumes that the two distributions are symmetrical?

Does the the equivalence of Total variation distance formulas presented here(https://ece.iisc.ac.in/~parimal/2019/statphy/lecture-14.pdf) assumes that the two distributions are symmetrical ?

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### A sufficient condition for the decomposition of a bounded random vector

Given a matrix $A \in \mathbb{R}_{+}^{n\times m}$, where $n<m$ and rank($A$)=n. Define a set (a zonotope) $\mathcal{C}(A)=\{(x_1,x_2,\ldots,x_n)|\sum_{i=1}^m{\bf{a}}_ix_i,x_i \in [-1,1]\}$, where ${...

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### A distribution $\pi \propto \exp(-f)$ satisfies log-Sobolev inequality, does $\exp(-af)$ also satisfy LSI?

Assume a distribution $\pi \propto e^{-f}$ satisfies log-Sobolev inequality (LSI)
$$\forall \rho \in P(\mathbb{R}^n), \quad KL(\rho\| \pi) \le \frac{1}{2\lambda} I(\rho \| \pi)$$
with LSI constant $\...

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### How close are two Gaussian random variables?

Given two Gaussian random variables A and B with (mean, standard deviation) of (a,s) and (b,m) respectively, is there a scalar w in [0,1] that indicates how close A and B are?

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### Moments related to a likelihood function

It is well known that, in the $n\rightarrow\infty$ limit, $$Q:=2\ln{L(\hat{\theta};\mathbf{X})}-2\ln{L(\theta;\mathbf{X})}$$ has the $\chi^2_1$ distribution, where $\theta$ is a parameter of a regular ...

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### How to recalculate the weights for an event that happens multiple times and requires all outcumes to be unique?

I think it's easiest to explain with an example.
I have a weighted probability list
A : 0.15
B : 0.15
C : 0.15
D : 0.1
E : 0.1
F : 0.1
G : 0.1
H : 0.075
I : 0.075
...

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### Literature on the phase shift of eigenvalues of a sample covariance matrix

Let $X_{p,n}$ be a random $(p \times n)$-matrix whose entries $(Y_{i,j})$ are iid. and $\mathbb{R}$-valued with the properties $\mathbb{E}[Y_{i,j}] = 0$ and $\mathbb{V}[Y_{i,j}]=1$. The sample ...

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### Asymptotic distribution of a specific ML estimator

Consider a random independent sample of size $n$ from a distribution defined by the following probability density function$$f(x)=\frac{3}{4\theta}\cdot\left(1-\frac{x^2}{\theta ^2}\right)$$ when $-\...

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93
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### The maximum trace of a covariance can be achieved by a discrete random vector?

Given a random variable $X$, satifying $P(0\leq X \leq 1)=1$, and $\mathsf{E}[X^2] = \alpha$. We know its maximum variance $\text{Var}(X) = \alpha(1-\alpha)$ achived by a binary random variable $P(X =...

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69
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### Transforming a Poisson distribution into a power law

Consider the probability mass function of the Poisson distribution given a mean $\lambda$:
\begin{equation}
\mathbb{P}\left(Y=k|\lambda\right)=\frac{e^{-\lambda} \lambda^{k}}{k !}
\end{equation}
By ...

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55
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### The moment problem for $m_n=1/n$

What is the p.d.f. for the moments $m_n=1/n$ ?
(They are obtained from $\int_0^1 x^n/x\ dx $, but clearly $1/x$ is not a p.d.f. on $[0,1]$)

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### Number of resampling until obtaining a uniform list

Let $A_0$ be a list of $ n$ distinct elements. By sampling with replacement the elements of $A_0$, we obtain a new list $A_1$ of $n$ elements that are not necessarily distinct. Repeat the same process ...

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### How does the integral of pseudo Gaussian density's partial derivative on $(0,\infty)$ depend on its variance?

Let $C_0$ be the set of continuous functions $a:[0,T]\to [0,1]$. Let $k: \mathbb R_+\times\mathbb R \to [1,2]$, $r:\mathbb R_+\to [0,1]$ be $1-$Lipschitz. For any $a\in C_0$, consider the pseudo ...

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### Non-affine smooth transformation of Gaussian is Gaussian

Suppose $Z\sim N(0,1)$ (standard Gaussian) and $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(Z)\sim N(0,1)$. My question is whether there exists any such $f$ other than $f(x)...

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### Probability that three vectors of a unit sphere lie on one side of a hyperplane if angle between the vectors are given

As the title says, How to find the probability of vectors a, b, c, on some unit sphere, all lies on same side of some hyperplane passing through the origin. Information present are the angles between ...

3
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1
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76
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### Questions on the integral of pseudo Gaussian kernel and its derivative on $(0,\infty)$

Consider pseudo Gaussian densities for $0<s<t$ and $x,y\in\mathbb R$
$$f(s,x,t,y):=\frac{1}{\sqrt{2\pi A(s,t,y)}}\exp\left(-\frac{(y-x)^2}{2A(s,t,y)}\right)\quad\mbox{and} \quad g(s,x,t,y):=\...

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1
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192
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### Another large noise limit

Note: Here all processes take values in $[0, 1]$.
Let $W$ be a standard one dimensional Brownian motion, and $\sigma > 0$ a constant.
Let $X$ be the solution to the SDE
$$dX_t = \sigma X_t \, dW_t$$...

4
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1
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### How does the integral of pseudo Gaussian kernel on $(0,\infty)$ depend on its variance?

Let $a, b: \mathbb R_+ \to [0,1]$ be continuous functions. Let $k: \mathbb R_+\times\mathbb R \to [1,2]$ be $1-$Lipschitz. Set, for $0<s<t$ and $y>0$,
$$A(s,t,y):=\int_s^t\frac{k(u,y)}{1+a(u)}...

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241
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### Does taking minimum preserve density monotonicity?

Suppose $X$ and $Y$ are continuous random variables with a joint density function $f_{X,Y}$. Both $X$ and $Y$ are supported on $(0,1)$ and have continuous (can be assumed differentiable) and non-...

4
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118
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### Existence of a joint distribution on Bernoulli variables with same probability of being pairwise different

Let $m\in\mathbb{N}$ and $p\in(0,1)$ be arbitrary. Is there a sequence $X_1,\dots,X_m$ of random variables with the following specs on their distribution:
Each $X_i$ is unbiased Bernoulli: $X_i\sim {\...

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1
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49
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### The distribution of the sum of values from a normal and a truncated normal distribution

Using R to extract truncated normal distribution samples and normal distribution samples separately, when they are combined, the image drawn by the hist function is very similar to a normal ...

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### Hammersley-Clifford theorem

The paper spatial interaction and the statistical analysis of lattice systems by Besag (1974) presents an alternative proof for the Hammersley-Clifford theorem.
In order to prove the HM theorem, Besag ...

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194
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### Prove or disprove a mutual information inequality

I have $n$ IID Bernoulli random variables denoted by $X_1,X_2,\ldots X_n$ with parameter $p$.
I am interested in knowing if the following inequality involving mutual information holds :
$\boxed{\max_{...

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1
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73
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### Mutual information between two discrete random variables

I have 2 IID random variables $X_1$ and $X_2$ with $Bern(p)$ distribution. I have another binary random variable $Y$ taking values in $\{0,1\}$.
I am interested in comparing the following 2 mutual ...

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### Monotonicity of alpha-stable CDF

It is a well-known fact that the bivariate CDF of a Gaussian random vector $(X, Y)$ with zero means, unit variance and correlation coefficient $\rho$ is strictly monotonically increasing in $\rho$, cf....

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73
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### Uniform distribution mod $1$ vs independence of random variables

Let $a_1, \cdots, a_k \in [0, 1)$ be real numbers such that $1, a_1, \cdots, a_k$ are independent over the rational numbers. By the Weyl equidistribution criterion in $k$-dimensions, we know that the ...

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1
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99
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### Compute inverse cdf of normal distribution [closed]

How can I compute inverse CDF of normal distribution using the central limit theorem on uniform distribution (u[0,1])

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### Сonditional characteristics with respect to a discrete random variable [closed]

160 asymmetrical coins participate in the first roll. In the second roll, only those coins on which the "eagle" fell out in the first roll participate. It is known that the probability of an ...

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### Closed-form CDF for bivariate normal distribution in point $(\Phi^{-1}(p),\,\Phi^{-1}(p))$

Let $\Phi(x)$ be a CDF of standard normal distribution and $\Phi^{-1}(p),\,p\in(0,1)$ its inverse.
It is evident that
$$
\mathbb{P}(X<\Phi^{-1}(p))=\Phi(\Phi^{-1}(p))=p,
$$
where $X\sim N(0,1)$.
Is ...

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0
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### Strong Rayleigh measures conditioned on partial sum

Consider a binary random vector $X=(X_1,\ldots,X_n)$ with a strong Rayleigh distribution (i.e., its multi-affine generating polynomial is stable). It is well known that the law of $X$ remains strong ...

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### Poisson distribution and conditional expected value [closed]

I have a task:
Lat's take independent variables $X_{i}$ with Poisson distribution $Poiss(a)$. Distribution of $a$ has density $p(a)=\frac{8}{3}a^3e^{-2a}, a\ge0$.
Calculate:
$E(a|X_1=3, X_2=2, X_3=5, ...

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### Show that $\mathbb{P}[ a V\le Z| V+Z]=\mathbb{P}[aV \ge Z| V+Z] \text{ a.s.} $ iff $V=\frac{1}{\sqrt{a}}Z'$ where $Z'$ is standard normal

Consider a pair of independent random variables $(V,Z)$ where $Z$ is standard normal. Now suppose that the following equality holds: for a given $a>0$
\begin{align}
\mathbb{P}[ a V\le Z| V+Z]=\...

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1
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81
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### Distribution of weight of special type of random-matrix vector product?

Let $G$ be a matrix of dimension $k \times n$ sampled uniformly randomly from $F_2^{k \times n}$. It is a well known fact that $y = xG$ is uniformly distributed in $F_2^n - \{0\}$ for all $x \in F_2^k$...

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1
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79
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### Random variable is Big O in probability notation

Let $R_n$ be a random variable with values in $[0,1]$ and $nR_n$ converges to $\frac{1}{1+C} \chi_m^2$ in distribution for some constant $C>0$ and $m\in \mathbb{N}$. Is it possible to show that $(1-...

3
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1
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116
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### Concentration inequality for Hilbert space valued random variables

I have read in a paper about the following result:
Let $V$ be a separable Hilbert space and $(\Omega,A_{\Omega},P)$ a probability space. Suppose that $Y_1,Y_2,...$ is a sequence of independent $V$-...

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1
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91
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### Fisher Information of variance of difference between random variable and gaussian

I'm reading through the following paper: https://arxiv.org/pdf/0704.1751.pdf
I'm stuck in the middle of page 8, at the statement:
$$E[||S(X)-S^*(X)||^2] = J(X) - J(X^*)$$
Where $S(X)$ is the score of ...

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1
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137
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### Infimum of Fourier transform of singular measure

Let $\mu$ be a finite non negative singular measure on $\mathbf{R}^d$. I would like to know if there exists some result on the infimum of the absolute value of its Fourier Transform $$\hat{\mu}(t)=\...

2
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1
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83
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### Examples of "almost" Ahlfors regular measures

Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$ such that there are $c,C,d,D>0$ satisfying: for every $x \in \mathbb{R}^n$ and every $r>0$
$$
c r^d \leq \mu(B(x,r)) \leq Cr^D.
$$
Let'...

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1
answer

124
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### Expected value of a function of normal random variable

Suppose $X\sim \mathcal{N}(0,\sigma^2)$, find the expectation $\mathbb{E}\left[\frac{1}{(1+X^2)^a}\right]$ where $a$ is a fixed positive real number.
Is there an explicit formula for the above ...

6
votes

1
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250
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### Is this function monotonically increasing?

Suppose that $\boldsymbol{t}\sim \mathcal{N}(\boldsymbol{u};\boldsymbol{0},\boldsymbol{M})=f_{\boldsymbol{t}}(\boldsymbol{u})$, where $\boldsymbol{t}$ is a $N$-dimensional gaussian random vector, and
\...

0
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1
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70
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### Integral form of expectation with respect to complex random variables [closed]

Let $h$ be a random variable and $g(h)$ be a real-valued function of $h$.
We know that if h is a real-random variable then:
$E_h[g(h)] = \int_{-\infty}^{\infty} f(h) g(h) dh$ where f(h) is the PDF of ...

3
votes

1
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124
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### On the convexity of certain set of random vectors

Let ${\cal X}$ be the set of pairs of random variables $(X,Y)$ with finite expectations. For constant $c\in[0,1]$, define set
$$
\{(X,Y)\in{\cal X}:\exists a\geq 0, \, b\geq 0 \text{ such that } E[\...

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0
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33
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### Maximum likelihood estimator for power law with negative exponent

Background
I have data that roughly follows a power law with a negative exponent (up to a point; also, the parameters of the "fit" were just guesstimated by eye as a demonstration):
Now I ...

3
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1
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312
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### Does convolution of a probability distribution with itself converge to its mean?

Suppose we have a probability density function $f(x)$ with a finite support $[a,b]$. If we take the probability convolution of $\lambda f $ with $(1-\lambda)f,0 <\lambda<1$ recursively for many ...

5
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2
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### Limit of the extremal process of i.i.d. Gaussians see from the tip

I'm trying to calculate the weak limit of $\mathcal{E}_N(x)=\sum_{k=1}^{2^N}\delta_{x-Z_k}$ , with $Z_k=X_k-\max_{k\leq 2^N}X_k$, $\{X_k\}$ being $2^N$ copies of i.i.d. Gaussians with mean zero and ...