All Questions
Tagged with convolution normal-distribution
62
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11
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Is there an exact expression for the full width half maximum of a sech^2 curve convolved on itself?
As some simple math can show, a Gaussian convolved onto itself is also a Gaussian. Importantly, the FWHM of the original gaussian compared to that of its convolved counterpart is different by a factor ...
- 11
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12
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Convolution of slightly multivariate Gaussians slightly modified
Starting with
$ p(a) = \int p(a|b) p(b) db$
replace $p(b)$ with $\tilde{p}(b) = \mathcal{N}(b; \mu_b, \Sigma_b + \tilde{D})$
where $\tilde{D}$ is an additive diagonal covariance.
Assuming ...
- 121
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36
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Preservation of strict log-concavity under convolution
I have spent an embarrassing amount of time trying to prove or disprove any of this. I am aware that a similar question was posted in 2014, but since I couldn't make anything out of the two sources ...
- 13
1
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43
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Every $f\in C_c^0$ is analytic?! Where is the mistake in the proof?
Let $f\in C_c^0(\mathbb{R})$ (continuous with compact support) and denote by $\rho_a$ the Gaussian density with standard deviation a and mean 0 (i.e. $\rho_a(x)=\frac{1}{\sqrt{2\pi a^2}} \exp({-\frac{...
- 21
1
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1
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68
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"Deconvoluting" the sum of independent random variables
I encountered the following question in my research. See here for some background, but this post should be self-contained. Suppose
$X$ follows a discrete uniform distribution on $m$ points $\{X_1, ...
- 836
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157
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Convolution of uniform and normal distributions
Consider a random variable U that has a uniform distribution on (0,1) and a random variable $X$
that has a standard normal distribution. Assume that $U$ and $X$ are independent. Determine an ...
- 3
1
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0
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22
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Given a multilinear polynomial $g$, and an odd function $f$ show that $\int f(x_i) \exp(-\| x \|^2) g(x-y) d x =0, \forall y,i$ where $g(x)=c$)
Suppose that the following convolution equation holds: for all $y\in \mathbb{R}^n$ and every $i =1 ..n$
$$\int f(x_i) \exp \left(- \frac{\| x \|^2}{2} \right) g(x-y) d x =0$$
where
$f(x)$ is an ...
- 6,015
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1
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39
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Inequalities with a normal convolution
Let $\mathbb{P}$ be any probability measure concentrated on an interval $[-1, 1]$. Why is it that for $\varphi$, the standard normal density, the following holds?
$$ (\varphi * \mathbb{P}) (x) = \int_{...
- 177
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25
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Calculating the Jacobian with respect to pixel intensity for Gaussian Blur
I'm applying a Gaussian blur to an image where each pixel of the original image is being optimized for some purpose and the Gaussian blur is an intermediate transformation on the pixel intensities. It ...
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2
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2
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124
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What is the absolute expected value of the sum of a distribution made up of Bernoulli and Normal distributions?
$$X \sim \mathcal N(\mu, \sigma^2)$$
$$Y \sim Bern(p)$$
$$Z = XY$$
I then have that the pdf of $Z$ is:
$$f(z) = (p-1)\delta(z) + p g(z)$$
where $g(z)$ is the pdf of the normal distribution.
I then ...
- 21
0
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0
answers
18
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Positivity of the gaussian convolution at the root
I am currently struggling on the following problem. Let $\sigma>0$ a real number and $g_\sigma$ the gaussian function with mean 0 and deviation $\sigma$. Let also $f$ be a smooth real function such ...
- 506
1
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93
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Computing $I=\int_{-\infty}^\infty \frac{e^{-\frac{1}{2}\left(\frac{x-\mu)}{\sigma} \right)^2}}{\sigma \sqrt{2 \pi}}\frac{1}{1+e^{-x}}\, \mathrm{d}x$
I had posted this on Stats Stackexchange as I initially thought a probabilistic approach would be best suited. But posting it here for as it has more traffic.
Problem
Evaluate $$I=\int_{-\infty}^\...
- 407
2
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87
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What is the probability distribution p that maximizes the overlap between the distribution p and its Gaussian convolution? [closed]
The question is for a given variance $\sigma_0^2$, to find a probability distribution $p(x)$, whose variance is $\sigma_0^2$, that maximizes the overlap between the distribution and its Gaussian ...
- 29
1
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1
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60
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Covariance of white noise smoothed by convolution with a squared exponential kernel
Determine the autocovariance
$$
C(s,t) = \text{Cov}(X(s), X(t))
$$
of white noise $W$ convolved with a squared exponential (Gaussian) kernel $\phi$
$$
X(t) = (\phi* W) (t) = \int \phi(t-x) W(dx)
$$...
- 2,435
2
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32
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Convolution of two functions doesn't fit my data as I thought it would [closed]
I have simulated a Gaussian curve in 50 bins of data. I have then repeated this many times, drawing the amplitude of the Gaussian from a log-normal distribution. Here are a 10 realizations:
(IMAGE 1)
...
- 149