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Questions tagged [dirac-delta]

Dirac delta is a generalized function, or distribution, on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line.

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Dirac delta function for multivariate input (in the context of Gaussian processes)

Let's say we have a set of $N$ observations $D = \{\bf X, t\}$ where ${\bf X} = [{\bf x}_1, ..., {\bf x}_N]^T$ are the locations and ${\bf t} = [t_1, ..., t_N]^T$ are the targets. When applying a ...
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spatial correlation function, power spectral density function and homogeneity condition

Suppose that the spatial correlation function $R_{VV}(\mathbf{x})$ of a zero-mean random field $V(\mathbf{x})$ ($\mathbf{x} \in \mathbb{R}^3$) and its power spectral density function $\widehat{R_{VV}}(...
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How is $s^k$ in p.g.f. likened to Dirac measure $\delta_k$?

How is $s^k$ (sometimes $z^k$, $\theta^k$) in probability generating function likened to Dirac measure $\delta_k$? I don't see how these are similar. Yet I see them equated. Clearly the Dirac ...
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Joint conditional density of two iid exponential random variables conditioned on their sum

I have the following question: Suppose $X_1, X_2$ iid $\sim f_X(x)=\theta e^{-\theta x}1\{x\ge 0\}$ and $S=X_1+X_2$. What is $f_{X_1,X_2}(x_1,x_2|S=s)$? The solution from our exercise class to this ...
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Parzen density estimates convergence

I am trying to understand why $lim_{||u|| \rightarrow+\infty}{\varphi(u)}\prod_{i=1}^{d}u_{i} = 0$ is necessary for convergence of Parzen density estimates. Similar question has been asked here ...
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Applying delta function for summing a discrete value [duplicate]

Can anyone give me an example how can I apply delta function for summing a discrete value (not discrete random variable, any value) with an exponential random value (or any continous variable ) ?
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Expectation with delta function

In Bishop on page 219, the equality below is given. Unfortunately, I fail to see how one finds the mean $\mu_{a}$ of this distribution. I know the definition of expectation operator, but how does one ...
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Correlation function of Wiener process (brownian)

The following is stated in a text I am using: Consider the Wiener process (Brownian process) $W(t)$. The Wiener process has no derivative $\xi(t) := \frac{d W}{d t}$, reflecting the fact that it ...
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What is the KL divergence of distribution from Dirac delta?

The Kullback–Leibler (KL) divergence of two continuous distributions $P(x)$ and $Q(x)$ is defined as $$D_{KL}(P \mid\mid Q) = \int_{X} P(x) \log{\left[\frac{P(x)}{Q(x)}\right]} dx$$ How can one ...
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how to perform marginalization of posterior to find the predicative distribution for bayesian logistic regression

I want to know how to show that $$p(a) = \int \delta(a - w^T x) q(w) dw$$ is gaussian, where $q$ is gaussian and $x$ is fixed, and $\delta$ is the dirac function. Everything below is just some ...
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How to define general covariance with a dirac delta with correct units

I want to define the covariance of a variable, say $X$, which is the integral over $Y(t)$: $$ {\rm Cov}[X,X'] = \int\int {\rm Cov} [aY(t), a'Y(t')] dt dt'.$$ Just as an example, let the variable $...
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Empirical probability and Dirac distribution

According to Deep Learning p.65(Ian Goodfellow and Yoshua Bengio and Aaron Courville, available online): (...) This can be accomplished by defining PDF using the Dirac delta function $\delta(x)$: ...
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Expectation of Dirac function

I am having some trouble understanding how to derive the expectation of white noise. Assuming we have $W(x)$ which is a continuous white noise process with zero mean and unit variance, Thus $$cov\big(...
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Role of Dirac function in particle filters

Particle approximations to probability densities are often introduced as a weighted sum of Dirac functions $$p(x) \approx \sum_{i=1}^N \omega^i \delta(x-x^i)$$ with the weights $$\omega^i \propto \...
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Dirac Delta function Notation

I am trying to understand the delta function notation used to be express a monte carlo approximation of a probability distribution. The notation used in this (p10) is $\pi(x_{1:n}) = \frac{1}{N}\sum^...
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Should Dirac's delta function be regarded as a subclass of the Gaussian distribution?

In Wikidata it is possible to link probability distributions (like everything else) in an ontology, e.g., that the t-distribution is a subclass of the noncentral t-distribution, see, e.g., https://...
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Using Dirac Delta functions for estimating a probability distribution

I'm having some trouble understanding this slide. It's mentioned in the context of gaussian distributions. I sort of understand the Dirac delta "function". The main difficulty I'm having is with the ...
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indicator variable - dirac delta or step function

I am trying to solve the following equation, \begin{equation} = \int_{-\infty}^{\infty} \frac{1}{\sqrt{ (2\pi)^{k_{Y}} | \Sigma |}} \cdot \mathrm{exp} \{ -\frac{1}{2} (Y - Xm)^{T} \Sigma^{-1} (Y ...
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Dirac delta function in likelihood function

I have tried to understand this myself but what I have found on the internet so far has not helped. I have a likelihood function that for part of it has the following statement: d0 is the Dirac ...
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Characteristic function of the Dirac delta?

What is the characteristic function of the Dirac delta function? Is it $e^{i*0}=1$?
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Delta function in monte carlo sampling

I am confused by the dirac delta function in the formulation of monte carlo sampling. http://www.cs.ubc.ca/~arnaud/doucet_johansen_tutorialPF.pdf, for instance, defines in section 3.1 page 8 the ...
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Mutual information with a Dirac delta type pdf

What does the $MI(X,Y)$ convey about $Y$, when one of the probability distributions, $X$ is trivial and has all the probability concentrated at a single point?