Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schartz's sense) or measures.

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Autoconvolution Notation

Is there any special notation for a list (or function) convolved with itself? For example, the convolution of the list [1,1,1,1] with itself produces a new list: [1, 1, 1, 1] * [1, 1, 1, 1] = [1,...
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31 views

Function equation with sth like self-convolution

I would like to get the solution of $\int_{x}^{y}f(t,y)f(x,t)dt=af(x,y)$ for all $\{y>x>0\}$, satisfying $\int _{0}^{x}f(t,x)dt=1$. Or a more simple question may be if we assume $f(x,y)=g(x/y)/...
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3answers
30 views

Find Laplace inverse

Let $${{{{x^{\ast}(s) = \left( \frac{1}{(s+\mu_1 + \mu_2) (s + \hat{\lambda}_2) (s + \lambda_1 +\lambda_2 )}\right)}}}}$$ be the laplace transform in question, where $\mu_1,\mu_2, \lambda_1,\lambda_2, ...
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1answer
15 views

Explaining Disjointness of Events

Below is a copy and paste of Dartmouth's explanation of the sum of independent random variables: Suppose $X$ and $Y$ are two independent discrete random variables with distribution functions $m_1(x)...
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1answer
134 views

Solving forced undamped vibration using Laplace transforms

I'm heaving trouble solving the following undamped forced vibration problem using Laplace transforms: $$\ddot{q}(t) + \omega_n^2 q(t) = \cos(\omega t).$$ I will show what I have done so far, and I'd ...
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26 views

Any way to simplify this integral?

$$\int_{-\infty}^{\infty}e^{-y\left(x\right)}\left(\log\int_{-\infty}^{\infty}e^{-y\left(z\right)}N\left(x-z,v\right)dz\right)dx $$ where $y(x)\equiv\sum_{i=1}^{n}c_{i}x^{i}$, $n$ is even and ...
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1answer
31 views

Support of convolution in pde

Let $f$ be an square-integrable function on a bounded domain U. Moreover $\eta(x) = C e^{-\frac {1}{1-x^2}} $ on $[-1,1] $ and 0 elsewhere. C is taken in such a way that the integral over eta is ...
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1answer
141 views

sum of two independent variables Z=X+Y, find pdf for Z

There are two independent random variables. Find density function of Z=X+Y $\mathbf{\\ f_X(x)=\begin{cases} 1- \frac{x}{2} & 0\leq x\leq 2\\ 0 & \text{otherwise} \end{cases}} \ \hspace{20pt}...
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0answers
46 views

How can I find the convolution of these two functions?

Restrucutred this question, as it felt more natural to be asked in a different matter than I first wrote. Given the functions $$f(t) = \frac{e^{-2|t|}}{4},\quad g(t) = \frac{e^{-3|t|}}{6}$$ Find ...
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2answers
48 views

Extension of the convolution theorem

From the convolution theorem, we know that the multiplication in the frequency domain is equivalent to convolution in the time domain, and vice-versa. I am wondering if there is some kind of ...
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83 views

sum of n i.i.d. random variables with a 2 parameter exponential distribution

I know that the sum of n random variables with an exponential(λ) distribution is distributed gamma(n,λ). But what is the gamma distribution formula for a sum of n random variables with an exponential(...
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38 views

How to evaluate Integro-Differential Equation using Laplace convolution?

Can someone please explain how I begin to evaluate the following integro-differential equation? I know that it involves a convolution, but the $y(τ)$ within the integral is throwing me off. $$\int_0^...
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0answers
89 views

Convolution between a discrete stochastic signal and a continuous function

I'm trying to find the convolution between a discrete, stochastic signal (for which I have data at each t) and an exponential decay function (which is continuous in t). Now I know that one can ...
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4answers
393 views

Finding convolution of exponential and uniform distribution- how to set integral limits?

I am studying the convolution method for creating the density function of two independent random variables and I am struggling with understanding how the bounds for integrals are created. There is ...
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1answer
26 views

How to calculate convolution with logarithm numerically?

I'm trying to compute an optimisation problem, which has a cost function involving $$I=\int_0^1\log|x-y|\rho(y)dy$$ where $x\in[0,1]$ and $\rho$ is a probability density. Eventually, I will want to ...
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2answers
402 views

Number Theoretic Transform (NTT) example not working out

I'm reading up on the NTT, which is a generalisation of the DFT. I'm working in $\mathbb{F}_5$ with primitive root $w=2 \mod 5$. Suppose I want to compute the NTT of $x=(1,4)$. So far I have obtained: ...
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0answers
22 views

How to sketch commutative property of convolution

I've been asked to sketch why $$y(n)=\sum_{k=0}^{\infty}x(k)h(n-k) = \sum_{k=0}^{\infty}h(k)x(n-k)$$ are equivalent expressions for convolution. I could explain this pretty easily with a proof, ...
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1answer
287 views

$X$ Poisson distribution, $Y$ geometric distribution - how to find $P(Y>X)$?

Suppose that $X$ has Poisson distribution with parameter $\lambda$ and that $Y$ has geometric distribution with parameter $p$ and is independent of $X$. What is the $P(Y>X)$ ? (the final formula ...
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0answers
46 views

How to approximate damped harmonic under random forcing

Given motion in response to random forcing, can a "simple" function be used to estimate future motion. Starting with the damped harmonic oscillator, we can write $${d^2 \over {dt^2}} h(t) + 2 \zeta \...
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2answers
1k views

probability mass function of sum of two independent geometric random variables

How could it be proved that the probability mass function of X + Y, where X and Y are independent random variables each geometrically distributed with parameter p; i.e. $p_X(n)=p_Y(n)=\left\{\begin{...
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1answer
219 views

Calculation of the convolution of Cauchy density function $\int_{-\infty}^{\infty}\frac{ab}{\pi^2}\frac{1}{y^2+a^2}\frac{1}{(x-y)^2+b^2}dy$

I tried to calculate the following integral, which is the convolution of Cauchy density function: $$\int_{-\infty}^{\infty}\frac{ab}{\pi^2}\frac{1}{y^2+a^2}\frac{1}{(x-y)^2+b^2}dy$$ I tried to use ...
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1answer
39 views

Inverse Laplace Transform of $F(s) = \frac{3s+8}{(s^2+2s+20)^2}$

Having a little trouble solving this fractional inverse Laplace were the den. is a irreducible repeated factor $$F(s) = \frac{3s+8}{(s^2+2s+20)^2}$$ tried to apply partial fractions to it and i just ...
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1answer
144 views

Gaussian smoothing kernel with different sigma values

I am not a mathematician by training, so excuse my lack of vocabulary or the imprecision in my question. I have a 1D distribution that I need to convolute, using a Gaussian kernel. However, all the ...
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0answers
72 views

Find the CDF of the sum of the inverse square of n random normal numbers

Question If I have n independent random normal numbers denoted $X_i$ each with mean $\mu_i$ and variance $\sigma_i$ (for $i = 1 ... n$). For each $X_i$ I have a weighting factor $w_i$. What is the ...
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0answers
28 views

Convolution basic

I am stuck with this question. Need some help. Solve for $y(t)$: $$ \frac{d^2y}{dt^2} + 3\frac{dy}{dt} + 2y = 2*x(t)$$ Given, $x(t) = \cos(tu(t))$, where $u(t)$ is unit step function, and with ...
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1answer
65 views

Does infinite repeated convolution with the same normal distribution converge?

According to Wikipedia, This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution of two distributions is normal if and only if both are normal. ...
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1answer
105 views

Estimating the integral of convolution by a product of integrals

I'm reading the appendix of Evans' PDE book. One part of it says if $V$ is a compact subset of $\mathbb R^n$ and $\phi_\epsilon$ from $\mathbb R^n$ to $[0,\infty)$ has support inside $B(0,\epsilon)$. $...
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1answer
56 views

What is Convolution?

The definition of convolution is known as the integral of the product of two functions $$(f*g)(t)\int_{-\infty}^{\infty} f(t -\tau)g(\tau)\,\mathrm d\tau$$ But what does the product of the functions ...
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1answer
98 views

Simple Discrete Convolution Question

With the discrete step function $$ u[n] = \begin{cases} 1, & n \ge 0 \\ 0, & n < 0 \\ \end{cases} $$ And the output $y[n]$ defined as a discrete convolution of the input $x[n]$ ...
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2answers
32 views

Evaluate convolution integral

Can someone tell me if I am calculating this integral correctly.
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1answer
160 views

Convolution integral involving two Heaviside functions

I am having trouble solving the following integral involving two Heaviside functions, obtained from a Laplace transform convolution: $\Large \int_0^t \frac{\tau}{\sqrt{\tau^2 - \alpha^2}} H(\tau - \...
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0answers
29 views

A small doubt regarding a previously asked limit of convolution

Previously, I asked this question to the forum. Pointwise limit of convolution Now, a question in this regard is coming to my mind. Suppose, we don't have the integral; i.e. we have the ...
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1answer
122 views

Max and sum of random variables

I have a set of independent random variables $\{A_1, A_2, B_1, B_2\}$. All of them have the same distribution function $F(x)$. I want to find distribution function of a variable $C$, where $C=max(A_1 +...
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1answer
46 views

Pointwise limit of convolution

Suppose $\omega$ is the standard mollifier in $\mathbb R$. Then, let $\omega_{\epsilon} (x):= \frac{1}{\epsilon} \omega \left(\frac{x}{\epsilon}\right)$. For $0 < t_{1} < t_{2}$ the following ...
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1answer
48 views

Combined arrival rate

Let us suppose a scenario with two clients, $a$ and $b$, each one generating load at rate $\lambda_a$ and $\lambda_b$, respectively. The server receives the requests from both clients. What will be ...
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220 views

Normal() convolved to Exp(polynomial)?

Is there a general exact solution for a Normal distribution convolved to $e^{y(x)}$, where $y(x)$ is a polynomial? $$e^{y(x)}*N(x,v)=\int_{-\infty}^{\infty}e^{y(z)}N(x-z,v)dz$$ $$y(x)\equiv\sum_{n=0}^...
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1answer
48 views

If $\phi$ vanishes outside of $|x| < 2$, $\phi = 1$ where $|x| < 1$, prove/disprove $f * \phi$ is in $L^1$/$L^2$

Problem Statement: Let $\phi$ be a positive, smooth function. Suppose $\phi$ vanishes outside a compact subset of $\{x : |x| < 2\}$ and satisfies $\phi(x) = 1$ if $|x| < 1$. Let $f$ be a ...
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0answers
31 views

Convolution of independent but 'different' probability distributions

I have the following two probability distributions they relate to a particular ice-cream: ...
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35 views

Uniformly boundedness of convolutions

Assume $X$ is an absolutely continuous random variable with pdf $f:\mathbb{R}\to[0,\infty)$. Assume further there exists $M>0$ s.t. $|f(t)|\leq M \quad\forall t\in\mathbb{R}$. Let $X_1,\dots,X_n$ ...
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3answers
94 views

Adding two random variables with convolution

I am trying to understand the purpose of convolution of two probability functions. Also when it is appropriate to use the convolve function on two independent probability distributions. ...
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1answer
54 views

convolution of probability measures

What do we mean by convolution of measures? With example What is the difference between convolution of measures and convolution of functions? What is probability measure? Give an example of ...
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74 views

Derivative of mollification

This is in response to a claim made in the second line of the question here, namely: Given the standard mollifier $\eta$ and a locally integrable function $f:U \rightarrow \mathbb{R}^n$, by defining ...
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32 views

For which $F$ we have $F(f\ast g)= f\ast F(g)$ for all $f,g \in L^{1}$?

Young's inequality tells us that: $L^{1}\ast L^{p} \subset L^{p}, (1\leq p \leq \infty)$ My Question: What are examples of functions $F:L^{1}(\mathbb R)\to L^{p}(\mathbb R)$ with the property $...
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47 views

Contracting a sum of convoluted delta functions using plus/minus symbol?

So I'm currently reading a journal article (I must say, I'm not too savvy with math notation). However, I'm coming across the following relationship: http://i.stack.imgur.com/omwcN.png The equation ...
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23 views

Prooving that multiply by exponent in time domain yields a frequency shift in frequency domain using convolution.

im trying to proove that $F[x(t)e^{-jat}] = X(w-a)$ using convolution. using the convolution property i know i should get a convolution of $F(x(t))$ and $F(e^{-jat})$ So: $$ F[x(t)e^{-jat}]= 1/2\pi ...
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1answer
57 views

Discrete convolution of two sequences

Let $R$ be a commutative ring with unity. A finite sequence $x=\left< x_0,\dots,x_n\right>$ with elements in $R$ is called to be prime if there exists $a_0,\dots,a_n \in R$ such that $\sum_{i=0}^...
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0answers
40 views

Convolution and space-time Fourier transform

I have a general function $u(x,y,z,t)$. Then, (1) what would be the space-time Fourier transform of $$G \star \frac{\partial^n u}{ \partial t^n }$$ and (2) would the relation $$G \star \frac{\...
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1answer
61 views

Convolution can smooth an input function, is there an operation which bunches it up?

An easy to remember description of what the convolution of two functions is, is to say that one is a weight function and the result is a weighted average of the other function. The canonical example ...
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46 views

Clarification on Wolfram Mathworld's explanation of the connection between Gelfand Transform and Fourier Transform

http://mathworld.wolfram.com/GelfandTransform.html In the definition, what does $x$, $\hat x(\phi)$, and $\phi$ represent exactly if we were to consider definition of the Fourier transform? Can ...
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1answer
55 views

Convolution vanishes on an interval

Fix a "test" function $f(x)=x\exp(-x^2)$, which is nonzero except $x=0$. Suppose that $g$ is a function with some necessary regularity. Consider the convolution. $$ (f\ast g )(x)=\int_{-\infty}^{+\...