Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [distribution-theory]

Use this tag for questions about distributions (or generalized functions). For questions about "probability distributions", use (probability-distributions). For questions about distributions as sub-bundles of a vector bundle, use (differential-geometry).

0
votes
0answers
13 views

Transforming probability function ,expected value

Given the following problem where I have to calculate the expected value of a given $T(x) = \frac{x^2}{2}\le8$ function. During the class we converted the distribution into uniform distribution so ...
1
vote
1answer
24 views

If $\varphi\in C_c^\infty(\mathbb R)$, then $\left|\varphi\right|\le\sup_{x\in\mathbb R}\left|\varphi'(x)\right|$

Let $\varphi\in C_c^\infty(\mathbb R)$ and $K:=\operatorname{supp}\varphi$. By the mean value theorem, $$\forall-\infty<a<b<\infty:\exists c\in(a,b):\varphi'(c)=\frac{\varphi(b)-\varphi(a)}{b-...
1
vote
1answer
14 views

Equating distributional derivatives

I'm trying to get unstuck from some problems I encountered while studying the Fourier transform on tempered distributions. I'll discuss them using the exercise that originated them. Let $\Lambda=\...
0
votes
1answer
24 views

definite integration - solution breakdown

$$ \int_0^{\infty} xe^{-x(y+1)}dx$$ $$=-\frac x {y+1}e^{-x(y+1)}|_0^{\infty} +\frac 1 {y+1} \int_0^{\infty}e^{-x(y+1)}dx$$ $$=\frac 1{(y+1)^{2}}.$$ Unfortunately i cannot follow the steps. I guess ...
-1
votes
0answers
20 views

If $N\to \infty $ why $n(x)=\frac{1}{N}\sum_{i=1}^N \delta (x-x_i)$ can be seen as a smooth function? [on hold]

Let $\{x_1,...,x_N\}$ a collection of real number (in fact the eigen value of a matrix). If $N\to \infty $ why $n(x)=\frac{1}{N}\sum_{i=1}^N \delta (x-x_i)$ can be seen as a smooth function in $x$ ? ...
0
votes
1answer
13 views

finding marginal distribution - how to determine the limits of integration

This exercise comes from Rice 3.12: let $f_{XY}(x,y)=c(x^2-y^2)e^{-x}, 0\leq x <\infty, -x \leq y \leq x$ b) find the marginal densities I have found thatc $c=\frac18$ and also that $f_X(x)=\...
0
votes
1answer
46 views

Find weak form of linear transport equation

I am stuck on the following problem that says: a) Find a weak formulation for the partial differential equation $${\partial u\over\partial t\ }+ c{\partial u \over \partial x\ }=0$$ b) Show ...
0
votes
1answer
23 views

A sequence of test functions that converges to a charscteristic function from beiow.

Let $E$ be a Borel bounded set of $\mathbb{R}^{n}$ and $\chi$ be the characteristic function of $E$. How would you construct a sequence $ \chi_{n}$ of nonnegative test functions bounded above by $\...
0
votes
1answer
36 views

Is the restriction of a test function still a test function?

Let $D$ be an open set of $\mathbb{R}^{m}$ with $m\geq1$. Suppose $\phi\in C_{c}^{\infty}(D)$ and $F$ is a closed set included to $D$. Can we say that the restriction of $\phi$ to $D\setminus F$ is in ...
1
vote
1answer
31 views

Let $T$ be exponential with parameter $λ.$ Let $X$ be discrete defined by $X=k,$ if $k≤T<k+1,$ $k=0,1,2,\dots$. Find the pdf of $X.$

I am aware that this question has been asked already here, however there is no accepted answer to it yet. I have no idea where to start. we know that $$ f(t)= \lambda \times exp (-\lambda t)$$ ...
2
votes
0answers
24 views

Discontinuous function extended as a distribution

Let $f:\mathbb{R}^\star\to\mathbb{C}$ be a continuous function on its domain, and suppose : $$\exists m\in\mathbb{N}^\star,\;\exists c>0\;/\;\forall x\in[-1,1]\setminus\lbrace0\rbrace,\;|f(x)|\...
0
votes
0answers
36 views

Two questions about mollifiers

Let $f$ be a smooth function whose laplacian equals 1 everywhere in $\mathbb{R}^{p}$ and $p>1$. Let $B(r)$ be the ball of radius $r>0$ and centered at the origin in $\mathbb{R}^{p}$, and let $ \...
2
votes
0answers
44 views

Distributional second-order derivatives of $\frac{e^{-|x|}}{4\pi |x|}$ to show the solution of $u -\Delta u=f$ is in $H^2$

In Brezis's book "Functional Anlaysis" it is proven that the solutions of the Helmotz equation $u - \Delta u=f$ where $f \in L^2 (\mathbb{\Omega})$ belong to $H^2 (\mathbb{\Omega}) \cap H^1 _0 (\Omega)...
1
vote
0answers
37 views

compactly support function with constant value

I am interested in finding a function $\varphi\colon\mathbb{R}\to\mathbb{R}$ with the following property: $\varphi$ has compact support, which contains $\left[0,1\right]$ $\varphi$ is continuously ...
1
vote
2answers
45 views

distribution associated with a discontinuous function

Let $f\colon\mathbb{R}\to\mathbb{R}$ be such that, for every $n\in\mathbb{Z}$, $f$ is differentiable on $\left(n,n+1\right)$ and $n$ is a discontinuity of first kind of $f$. We define $$T_f(\phi)=\...
1
vote
1answer
39 views

A basis in the space of all tempered distributions over R^n

What is a(n uncountable) basis in the topological vector space $\mathcal{S}' \left(\mathbb{R}^n\right)$ ? How can any tempered distribution be expanded in terms of such a basis?
-1
votes
1answer
25 views

How to calculate expected value and variance of a random variable [closed]

Let the random variable $Y$ have the following density: $$f(y) = \frac{1+\beta y}2, -1 \le y \le 1, -1 \le \beta \le 1$$ Find $E(Y)$ and $V(Y)$. Can anyone help me with this problem?
2
votes
1answer
60 views

Convolution with Gaussian

Let $f, g\in \mathcal{S}(\mathbb R)$ (Schwartz class function), $\delta_0$ (dirac delta distribution). Consider distribution as follows: $$G(x, y)= f(x)g(x)\delta_0(y)-f(y)g(y)\delta_0(x), \ (x, y\...
0
votes
0answers
35 views

Kind of passage to the limit in the sense of distributions

Suppose $B$ is a ball in $\mathbb{R}^{n}$ with $n>1$, and $f$ a locally integrable function. Suppose $F$ is a closed set with empty interior and $M>0$ such that the distribution defined by $f$ ...
2
votes
0answers
26 views

Dirac distribution δ(x-x)

When canonically quantising a scalar field without normal ordering, one comes across the following expression: $$\int_\mathbb{R}E(x)\delta(x-x)dx.$$ Here, $E(x)$ is a smooth unbound function. I have ...
0
votes
1answer
21 views

No convolution Identity element in $L^1_{per} $ using Fourier series

We have to show that there is no identity element for the ring $ L^1_{per}( ]0,2\pi [) $ specifically using the Fourier coeffcients. Suppose that : $ \exists e , \: \: e * f = f \: \: \: \: \forall f ...
0
votes
0answers
15 views

What is the distribution of the integral of GBM on a finite support?

From this topic: Power of the integral of a Geometric Brownian motion I know that the random variable: $$ X = \int_0^\infty e^{aB_t-bt} dt $$ has the Inverse-Gamma distribution with some parameters (I ...
0
votes
0answers
11 views

Integral of inverse fuctions for noncontinuous distributions

Let $ F:[a,b]\rightarrow[0,1] $ be an arbitrary (noncontinuous) distribution function. Denote with $Q(p)=inf\{x:p \leq F(x)\}$ the associated quantile function. I would like to use that $ \...
0
votes
1answer
25 views

A function in $\textit{D}(\mathbb{R})$

I was solving an exercise and I faced these 2 parts 1) Prove that there exist a function $\gamma \in D(\mathbb{R})$ such that $\gamma(0)=0$ and $\gamma'(0)=1$. I tried for this part the function $$\...
0
votes
0answers
25 views

Compactly supported functions

Given $$g=g(x):\mathbb{R}\to \mathbb{R}$$ $g\in C^1(\mathbb{R})$ with compact support, Is It always possibile to construct $v=v(t,x) \in C^1([0,+\infty)\times\mathbb{R})$ such that $v$ has compact ...
0
votes
0answers
7 views

Distribution of the following quantity after Gram Schmidt Orthogonalisation

Suppose $X_1,X_2,\dots,X_n$ are p-dimensional random variables with distribution $N_p(\mu,\Sigma)$. Let $X_{n\times p}$ is the data matrix. $\mathbb{Z}_{n\times p}= (n-1)^{-1/2}\left(I_n - n^{-1}\...
2
votes
1answer
65 views

Spherical Laplacians on an Exponential

I looked around a bit and couldn't find a resolution to this. I was curious about the scalar function $u(r) = e^{-r}$ with $r \in [0,\infty)$ and acting Spherical Laplacians on it. $$\Delta u(r) = \...
0
votes
0answers
24 views

Reference : Regular open, local proof

Look at this proof ! ce This is a powerfull way of thinking, it must come from a more general theory. The way of process is too powerfull to be just an aspect of Sobolev space. Do you know any ...
2
votes
2answers
101 views

What is a distribution ? Why it's not really a function?

In a book I'm reading (functional analysis of Stein and Shakarshi), they say that a distribution $F$ is not be given by assigning value of $F$ at most point, but will instead be determinated by its ...
0
votes
1answer
29 views

Approximating the delta distribution

The delta function (well, delta is not really a function; it is a distribution) can be defined as a limit of (among many other approximations) the following approximates of the unity: the heat kernel:...
2
votes
0answers
50 views

Hidden Fourier multiplier in integral expression?

After some (formal!) manipulations I stumbled upon the following expression: $$ \hat{f}\left(\xi,\eta\right)=\iint_{\mathbb{R}^{2n}}e^{2\pi i\left\langle x,t-\xi\right\rangle }e^{2\pi i\left\langle \...
-1
votes
0answers
20 views

Find the distribution of the pooled variance estimator

The two-sided 95% confidence interval of $\mu_x$ based on 10 independent (inaccurate) measures X of the distance (in metres) between a point A and a point B with a certain measuring instrument is $$[...
0
votes
1answer
21 views

transformation and marginalization of a joint distribution function

Suppose that the joint probability density function of X1 and X2 is given by: $$f(x_1; x_2) = exp(−x_1 − x_2)$$ $x_1 > 0$; $x_2 > 0$ and $0$ $elsewhere$. We define Y1 and Y2 as follows: $$Y_1 = ...
2
votes
1answer
26 views

Solve $xu = 0$ in the sense of distribution

I got stuck at showing $\delta_0$ solves $xu=0$ in the sense of distribution (up to some constant). The hint states to decompose $\phi = \phi(0)g(x)+x\varphi(x)$ for some function $g(x),\varphi(x) \...
0
votes
1answer
36 views

Delta derivative distribution identity?

It is easy to show that the Dirac $\delta(x)$ distribution satisfies the distributional identity $$\delta(x) = - x \delta'(x).$$ Can we conclude that the following also holds $$\delta'(x) = - \frac{\...
1
vote
1answer
42 views

$\widehat{\frac{1}{x^3}}=C\,\xi^{2} sgn(\xi)$?

How understand this Fourier transform: $\widehat{\frac{1}{x^3}}=c \xi^{2} sgn(\xi)$ ? The function $x\mapsto\frac{1}{x}$ is not locally integrable, so it does not define a distribution through ...
1
vote
3answers
44 views

Convolution between two distributions $T, S \in \mathcal{D}'$

I search a book where it is explained how the convolution between two distributions $T, S \in \mathcal{D}'(\Omega)$ is defined. Thank you in advance.
2
votes
1answer
43 views

Are distributions all continuous?

By a distribution, I mean it is a linear functional of the space of smooth compactly supported functions over $\mathbb R^n$, i.e. $C_c^{\infty}(\mathbb R^n).$ I am reading a textbook by Strichartz, ...
2
votes
1answer
56 views

Necessary and sufficient condition for $a_n\delta_n\to0$ in $\mathcal{S}'$; example of sequence convergent in $\mathcal{D}'$ but not in $\mathcal{S}'$

The sequence $(a_n\delta_n)_{n\in\mathbb{N}}$, where, for each $n\in\mathbb{N}$, $a_n$ is a complex number and $\delta_n$ the Dirac delta translated of $n$, i.e. $\langle \delta_n,\phi\rangle=\phi(n)$ ...
0
votes
3answers
52 views

Is $\ln |x| \in L^1_{loc}$?

I want to prove it to define an distribution in $S'(\mathbb{R})$, but I don't know if $\ln|x|$ is in this space.
0
votes
0answers
17 views

Distributional derivative and Laplace-Beltrami operator

Let $\Omega \subset \mathbb R^3$ be an open, bounded and connected set with $C^2-$regular boundary $\Gamma$. Consider an arbitary test function $\phi \in C^2(\Gamma)$. If $\Delta$ here denotes the ...
1
vote
1answer
29 views

distribution-valued integral in Grafakos

In the book Modern Fourier Analysis by Grafakos in the proof of the T(1) Theorem in the implication that shows $L^2$ boundedness (in the 3rd edition this is on page 243), Grafakos considers a ...
0
votes
2answers
18 views

probability of ${Unif(0,1)^2}$

I am new to probability theory. In computer programming, I often use the uniform random number in $(0,1)$ $$ U= Unif(0,1) $$ what is the probability density of $U^2$? In general, how to find the ...
1
vote
0answers
35 views

Where do functions for which $\partial_x\partial_y \neq \partial_y \partial_x$ fit in the theory of distributions?

Consider $f$ defined on $\mathbb R^2$ by $$ f(x,y) = \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} & (x,y)\neq 0 \\ 0 & (x,y)= 0\end{cases}$$ It is known that $\partial_x \partial_y f \neq \...
1
vote
1answer
31 views

How exactly does the class of generalized functions include that of ordinary functions?

I just dabbled into the field of generalized functions (or theory of distributions). I am a bit confused as to how the class of generalized functions includes that of ordinary functions. Let $K$ be ...
0
votes
0answers
29 views

Show that if $u$ and $v$ are distributions and $u(\phi)=0$ for all $\phi\in D(\Omega)$ for which $v(\phi)=0$, then $u=cv$

Let $\Omega$ be a nonempty and open subset of $\Bbb R^n$. Show that if $u$ and $v$ are distributions on $\Bbb R^n$ and $u(\phi)=0$ for all $\phi\in D(\Omega)$ for which $v(\phi)=0$, then $u = cv$ for ...
0
votes
1answer
65 views

$\{e^{-x^2/n^2}\}$ converges to the generalized function 1

$\{e^{-x^2/n^2}\}$ converges to the generalized function 1 I could not prove this result from "M. J. Lighthill-Introduction to Fourier analysis and generalized functions (1964)" (p.17) It is easy to ...
1
vote
1answer
46 views

Calculation probability

I have the following exercise and I do not know if I have raised it well in the first 2 sections and section 3 I do not know how to start. A freight train loads containers of three types: (a) of 36 ...
1
vote
1answer
29 views

Fourier transformation of an $L^1$ function

Let the function $f$ be defined on $\mathbb{R}$ by $f(x)=e^{iax}$ with $a \in \mathbb{R}$. We have that $f \in S'(\mathbb{R})$, and $F \delta_a= e^{-iax}$ and $\overline{F} \delta_a=f$. where $F f $ ...
0
votes
1answer
26 views

Are the probability of observing extreme values in hypothesis testing independent?

Im trying to understand some definitions of the setup in two-sided hypothesis-testing. The probability of observing extreme values when doing two-tailed hypothesis-testing can be evaluated (if I'm not ...