# 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
1 vote
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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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### 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]=\...
1 vote
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 $x \in F_2^k$...