Questions tagged [poisson-distribution]
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14 questions
4
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Transforming a Poisson distribution into a power law
Consider the probability mass function of the Poisson distribution given a mean $\lambda$:
\begin{equation}
\mathbb{P}\left(Y=k|\lambda\right)=\frac{e^{-\lambda} \lambda^{k}}{k !}
\end{equation}
By ...
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3
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1
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376
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Random pseudo-walk of Poisson variables
Suppose there is a pool that can contain any non-negative number of objects. At time $t$ it contains $n_t$ objects. Time is discrete.
Before time $t+1$ two things happen, in this order:
Unless the ...
- 61
3
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2
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Is every discrete compound Poisson distribution a mixed Poisson distribution?
I asked and bountied this question at math SE but didn't get any answers, so I suspect that only experts (if anyone) may know the answer.
The mixed Poisson distribution and compound Poisson ...
- 1,311
2
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1
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Random pseudo-walk with 'disappearing' values
This question is a twist on a question I asked here Random pseudo-walk of Poisson variables, but with randomly 'disappearing' objects. I do not know how to generalize the (satisfactory) answer given ...
- 61
2
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0
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Poisson process subordinated by a gamma process
I am working on a problem and I encountered the following situation:
$(N(t): t \ge 0)$ is a Poisson process with parameter $\lambda t $. If $T_{n} = \sum_{i=1}^n W_i$ represents the $n^\text{th}$ ...
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2
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0
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126
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A question related to the jumps of a Levy process
The Lévy–Khintchine formula says that any Lévy process, $X=(X(t), t \geq 0)$, has a specific form for its characteristic function. More precisely, for all $t \geq 0$, $u \in \mathbb R^d$:
$$
\mathbb{E}...
- 121
1
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0
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Gamma and Poisson distributions and their relations to the randomness
I'm reading the following paper:
https://academic.oup.com/bioinformatics/article/32/1/122/1743683
and in Figure 3 (Section 4.4) the authors have shown some vertex degree distributions:
enter image ...
1
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1
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81
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The Lévy process jumps
I have two questions.
Let $(X_t)_{t\geq 0}$ be a Lévy process with Lévy measure $\nu$. The jump process $\Delta X=\left(\Delta X_t\right)_{t\geq 0}$ is defined by
$\Delta X_t=X_t-X_{t-}$, for every $t\...
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1
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0
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Given p1=P(X1>X2), p2=P(X1=X2) and p3=P(X1<X2) (with necessarily p1+p2+p3=1), what is λ1 and λ2？ [closed]
In the following three equations : p1, p2, p3 are known, find λ1=? , λ2=?
(with necessarily p1+p2+p3=1)
$$ p_1 = \sum_{0 \leq k_2 \lt k_1 \in Z}(\frac {\lambda _1^{k_1}} {k_1!}\cdot{e^{-\lambda _1}})(\...
- 29
0
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1
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Was this proposition on cumulants of compound Poisson distributions known before I put it into a Wikipedia article?
The $n$th cumulant $\kappa_n$ of a probability distribution for $n\ge2$ is functional that is a polynomial in the first $n$ moments of the distribution, that has the properties of $(1)$ homogeneity, $(...
0
votes
1
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301
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Poisson Process x SIR model [closed]
Consider the simplest SIR model:
$$S'=-a SI$$
$$I'=a SI - b I$$
$$R'=b I$$
It is known that the waiting time of an infeccious person in the compartment $I$ follows an exponential behavior with rate $b$...
- 123
0
votes
1
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139
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Lévy measure and jump behaviour of the corresponding Lévy process
Let $(X_t)_{t \ge 0}$ be a Lévy process on $\mathbb R$ with Lévy measure $\nu$.
Define the jump counting measure $N(t, A) = \lvert\{s \in [0, t] \mathrel: \Delta X_s \in A\}\rvert$
where $\Delta X_s$ ...
- 77
0
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1
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Characteristic function of quadratic variation of compound Poisson process
If I have a compound Poisson process whose characteristic function is known, is there a way to calculate the joint characteristic function of this process and its quadratic variation process?
If not ...
0
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0
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Power expectation involving a Poisson process
Consider a Poisson process $N_t$ with intensity $\lambda>0$ and let $x$ be a real-valued number. In principle, from the properties of the Poisson distribution, we have:
\begin{align}
\mathbb{E}(x^{...
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