All Questions
Tagged with convolution probability-distributions
15 questions
6
votes
0
answers
305
views
Distribution class closed under convolution counterexample?
Define the class of probability density functions $\mathcal{C}$: $\,p \in \mathcal{C}$ iff $p(x)=p(|x|)$, and $\log p(\!\sqrt{x})$ is convex on $[0,\infty)$.
Conjecture: if $p,q \in \mathcal{C}$, then ...
- 391
0
votes
0
answers
65
views
Probability distribution of total time for a job, given a workflow graph
$$
\begin{array}{cccccccccccc}
& & \text{A} \\
& \swarrow & & \searrow \\
\text{B} & & & & \text{C} \\
& \searrow & & \swarrow \\
\downarrow & &...
0
votes
0
answers
44
views
Are there probability densities $\rho, f_n$ such that $\lim_n \frac{[\rho * f_n]_\alpha}{\|\rho * f_n\|_\infty} = \infty$?
We fix $\alpha \in (0, 1)$. Let $[f]_\alpha$ be the best $\alpha$-Hölder constant of $f: \mathbb R^d \to \mathbb R^k \otimes \mathbb R^m$, i.e., $[f]_\alpha := \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x-...
- 825
1
vote
1
answer
58
views
Lower bound the best $\alpha$-Hölder constant of a convolution
Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (...
- 825
1
vote
2
answers
90
views
Is the difference between $\alpha$-Hölder constants of $f*\rho$ and $g*\rho$ controlled by $\|f-g\|_\infty$?
Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (...
- 825
0
votes
1
answer
296
views
When can a convolution be written as a change of variables?
Suppose $X$ is a random variable with a density $f(x)$ such that $f(x)$ is a convolution of some density $g$ with some other density $q$:
$$
f = g\ast q.
$$
Under what conditions does $X=h(Y)$, where $...
- 5
2
votes
1
answer
338
views
Uniqueness of deconvolution after convolution?
I have the following question and I'd greatly appreciate any help!
Basically, I have an arbitrary probability distribution with pdf $f(x)$, we can assume it's continuous with support on $[0,\infty]$
...
- 31
7
votes
2
answers
880
views
Which random variables can be written as the difference of two independent positive random variables?
Can we characterize random variables $X$ that satisfy
$$
X\sim Y - Z
$$
for two independent positive random variables $Y$ and $Z$?
Are $Y$ and $Z$ unique in some sense?
Can (one possible choice of) $Y$...
- 302
1
vote
1
answer
2k
views
Convolution of two Gaussian mixture model
Suppose I have two independent random variables $X$, $Y$, each modeled by the Gaussian mixture model (GMM). That is,
$$
f(x)=\sum _{k=1}^K \pi _k \mathcal{N}\left(x|\mu _k,\sigma _k\right)
$$
$$
g(y)=\...
- 41
2
votes
1
answer
332
views
Prove or disprove the linearity of expectiles
For expectation (mean), there are many useful properties such as Linearity of Expectation:
$\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y]$
$\mathbb{E}[\alpha X]=\alpha\mathbb{E}[X]$
(The two equations ...
- 139
2
votes
1
answer
403
views
Product of independent random variables and tail deconvolution
Suppose $X, Y$ are two independent non-negative random variables. The conditions
(i) $\mathbb{P}(X > t) = \frac{C}{t^p} + o(t^{-p})$
(ii) $\mathbb{P}(Y > t) = o(t^{-q})$ for any $q > ...
- 205
1
vote
0
answers
47
views
Show a convolution of distributions ε-close to min-entropy k is ε^t-close to min-entropy k
Assume $X_1,...,X_t$ are independent distributions on $\mathbb{Z}_2^n$ s.t. each $X_i$ is $\epsilon$-close to min-entropy $k$; i.e. there exist distributions $Y_1,...,Y_t$ on $\mathbb{Z}_2^n$ s.t:
$$ \...
- 21
3
votes
2
answers
278
views
The disintegration of the convolution of two probability measures
Let $G$ be a topological group with all the topological conditions in order that some form of the disintegration theorem be applicable (for instance, take $G$ metrizable). Let $N$ be normal and closed,...
- 5,407
3
votes
0
answers
119
views
Any chance to get the moments of this exotic distribution?
Let us define the following cumulative distribution:
\begin{align}
\Pr (Y(t)\geq a)=\sum_{n=0}^\infty \frac{e^{-kt}(kt)^n }{n!}\int_a^\infty[\circledast_{f}\circ H_a ]^n\delta (x) dx
\end{align}
where ...
- 634
3
votes
0
answers
81
views
Computing distribution of non-identical coin flips
Suppose I have $N$ coins, where coin $i$ has probability $p_i$ of coming up heads. I flip all $N$ coins and let $S_N$ be the number of heads. How can I compute the distribution of $S_N$ efficiently?
...
- 3,979