All Questions
Tagged with convolution probability-theory
143
questions
2
votes
1
answer
79
views
Proving a distribution is not infinitely divisible
I'm trying to show the following:
Show that the distribution on $\mathbb R$ with density $f(x) = \frac{1-\cos(x)}{\pi x^2}$ is not infinitely divisible.
The characteristic function of this ...
- 4,075
0
votes
0
answers
35
views
Dependent random variables whose convolution adds up
I want to find two dependent random variables $X$ and $Y$ with values in $\mathbb{Z}$ such that $P_X\star P_Y=P_{X+Y}$, where $\star$ is the convolution.
What I've tried:
I was "tickling" ...
- 1,279
5
votes
2
answers
248
views
Calculate $\sum_{n=1}^\infty (1-\alpha) \alpha^nP^{*n}(x)$
$$
\mbox{Let}\quad
P'(x)=\sum_{j=1}^n a_j\frac{a^jx^{j-1}}{(j-1)!}e^{-ax},x\geq 0
$$
be the density function of a mixture of Erlangs and let $\alpha\in(0,1)$:
Is is possible to determine an analytic ...
- 91
0
votes
0
answers
15
views
Does there exist a convolution semigroup with compactly supported density?
I am looking for a convolution semigroup $(P_t)_{t \geq 0}$ of operators acting on $C_0^\infty(\mathbb{R^d})$, of the form
$$P_tf(x) = (p_t * f)(x) = \int_{\mathbb{R}^d} p_t(x-y) f(y) dy,$$
where $p_t$...
- 1,633
4
votes
1
answer
179
views
defective renewal equation
I am reading this paper of Lin and Willmot. I dont understand how they come up with formula $2.7$ and why $\tilde{\phi}(s)=\frac{\tilde{H}(s)}{1+\beta-\tilde{g}(s)}$. Can someone help me?
So i want to ...
- 351
1
vote
0
answers
39
views
Are there any interesting families of probability measures on $\mathbb{R}$ closed under convolution and "deconvolution"?
To define the property I am interested, let me introduce a notion of convolution measure on a measurable monoid $\mathcal{S}$. However, I am predominantly interested in the case where $\mathcal{S} \...
- 170k
1
vote
1
answer
79
views
Sum of frequency distributions vs convolutions
From my understanding, the sum of independent random variables will be the same as the convolution of the input distributions.
However, when experimenting with it, I see the distribution of the sum of ...
- 143
1
vote
0
answers
28
views
Question about step in proof of Young's convolution Inequality
This is the inequality we are attempting to prove: Let $f \in \mathcal{L}^p(\mathbb{R}^n), \ g \in \mathcal{L}^1(\mathbb{R}^n)$ and $1 \leq p < \infty$. Then $f * g$ is defined almost everywhere ...
- 11
1
vote
0
answers
46
views
Sum of two random variables (General Formula)
Rather than attempt to derive a general expression for the
distribution of $X + Y$ in the discrete case, we shall consider some
examples. (Ross, A First Course in Probability, Section 6.3.4)
The ...
- 1,834
0
votes
1
answer
111
views
Independence of sum of i.i.d. random variables from remaining variables
The following question is very similar to this reference.
Let $(X_{1},X_{2},X_{3})$ be a random vector with continuous independent variable entries. We denote by $X_{1}^{1}, X_{1}^{2}$ two variables ...
- 35
1
vote
1
answer
60
views
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
1
vote
1
answer
124
views
Compound Poisson Distribution and its Expected Value
I am trying to understand the proof of Theorem 16.14 of Probability Theory by A. Klenke (3rd version) about the Levy-Khinchin formula.
I would like to know how to prove this:
$$E[X]=\int x e^{-v(\...
- 563
0
votes
2
answers
131
views
How to compute the PDF of $Z=XY$
After the answer I got, I'm interested in the case for computing the PDF of a random variable $Z=XY$, and $X, Y$ are independent. The convolution of PDF describe the PDF of the sum of two random ...
- 495
4
votes
1
answer
391
views
How do I find the probability distributions for multiple dice rolls for dice with a differing number of sides?
I have been learning how to play Dungeons and Dragons recently, and have bought my first set of dice.
The standard DnD dice are:
1d4
1d6
1d8
1d10
1d10 × 10 [10, 20, 30... 90, 100]
1d12
1d20
I was ...
- 163
1
vote
0
answers
41
views
What is the practical reason in defining a convolution by different indeces
I am taking some time to carefully construct my understanding of generating functions as they are extremely interesting to me in terms of properties and uses.
I am making my way to convolutions now ...
