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Questions tagged [skellam-distribution]

The Skellam distribution is a discrete distribution that describes the difference between two independent Poisson distributions with possibly different parameters.

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What is the quantile function of a Skellam distribution?

Reading Is there a formula for the 95% confidence interval for the Skellam distribution or difference of two Poisson r.v.'s?, I realized that I would rather have the quantile function. Unfortunately ...
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What is the PMF of the Zero-Truncated Skellam distribution?

Vaguely related to the notion of a Lindley equation, I am considering a recurrence $$L_{t+1} := L_t + \max \left( 0, A_t - S_t \right)$$ where $$L_t$$ is the number of customers in a queue system at ...
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Should $3\!-\!$pointers be worth $2.5{\it ?}$

I was seeking an alternative scoring rule sets instead of three points for a win (gained more engaging and balanced) to cancel the theory "Banking a draw meaning a new kind of loss". It's ...
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Proof of $\left|c_{i}-c_{j}\right|\left\{\begin{matrix}\ge 2,{\rm of}\,3/4\,{\rm probability}\\<2,{\rm of}\,1/4\,{\rm probability}\end{matrix}\right.$

FA Premier League 2019/20. The season was affected by the COVID-19 Pandemic while each team had a so-called quarter of their schedule left. ("quarter" ? Since each team has 4/9 or 5/9 number ...
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MGF of the absolute Value of a Skellam RV

I am trying to derive the moment generating function for the absolute value of a Skellam random variable $Skellam(\lambda_1, \lambda_2)$ Suppose $X_1 \sim Pois(\lambda_1)$ and $X_2 \sim Pois(\lambda_2)...
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Numerical evaluation of infinite sums

I am working with Skellam random variables and I would like to evaluate the CDF of the absolute value of a Skellam random variable in which both Poisson random variables have the same rate, $\lambda_1 ...
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The convolution of a Poisson Distribution and Scaled Poisson Distribution

I am trying to do a likelihood analysis on a variable, $Z$, which is defined as $(1)$ $Z = X - cY$ where $X$ and $Y$ are both independent Poisson distributions with rate parameters $\lambda_{x}, \...
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Is it always possible to find $X\sim\text{Pois}(\lambda)$ and $Y\sim\text{Pois}(\mu)$ with given $P(X<Y)$, $P(X=Y)$ and $P(X>Y)$?

Consider $P(X<Y)$, $P(X=Y)$ and $P(X>Y)$ in $(0,1)$ that sum to $1$. Is it always possible to find Poisson parameters $\lambda$ and $\mu$ such that independent $X\sim\text{Pois}(\lambda)$ and $...
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Deriving the mean values of summed poisson distributions

z is a poisson distribution, in which λ = 2.887. It has been formed by summing two other independent poisson distributions, x and y. Three further pieces of information are given about the ...
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Simplifying modified Bessel function of the first kind

The modified Bessel function of the first kind shows up in the normalizing constant of a lot of random variables (e.g. the normal product distribution, the noncentral chi-square distribution, the ...
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What probability distribution would be suitable for modelling scores in a basketball match?

In football (not American football but what Americans call soccer), it is pretty clear: we may consider the difference of two independent Poisson variables. In basketball, in theory, we could increase ...
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What is the expectation of the absolute value of the Skellam distribution?

In particular, for a Skellam distribution obtained by substracting two iid Poisson Processes. Thank you!
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Is there a reason why the regression in the R Skellam package uses three optimisation steps?

I am not entirely sure whether this would be better posted in Stack Overflow, or maths.se, however as it is about model fitting I thought I would try here first. The Skellam distribution $\...
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Distribution of one Poisson variable relative to another

Assume A and B are two independent Poisson random numbers with means a and b. What is the conditional distribution of A given that A>B, A=B and A...
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What is the correct way to test if there are any statistically significant differences between values on some variable between any two years?

Let's say I have the murder rate for a large number of years. What would the correct way to test to see if there are any statistically significant differences between these rates for any two years? ...
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equivalent of t-test for Binomial/Poisson variables

I have to try to estimate and explain conversion rates that can be extremely low, on limited dataset. Because I have very few observations, a normal framework would give me a poor estimate, because ...
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Computing $P(Z>0)$ with $Z\sim\mathrm{Skellam}(k;\lambda_1;\lambda_2)$

Given the variable $Z$, distributed as follows: $Z\sim\mathrm{Skellam}(k;\lambda_1;\lambda_2)$ I would like to understand how to compute the probability $P$$(Z>0)$. I know to compute the value $...
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From the bivariate poisson to the Skellam (or Poisson Difference) distribution

I am looking at how to calculate the Skellam distribution (https://en.wikipedia.org/wiki/Skellam_distribution) from the bivariate poisson distribution. I understand that the Skellam comes from ...
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Testing if the difference between two count variables is different from zero

I have two count variables for several hundred thousand comparisons, one expected and one observed, and I would like to test if the counts are significantly different. One possible approach I have ...
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Skellam distribution with dummies - consistent ML estimates?

The difference of two Poisson distributed random variables follows a Skellam distribution. We know that a Maximum Likelihood estimation using Poisson distribution yields consistent estimates also in ...
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Does the noise term in a SDE need to be Gaussian?

Most of the examples I've seen for stochastic differential equations are of the form: $$ dX_t = \mu(X_t, t)dt + \sigma(X_t, t) dW_t $$ where $dW_t$ is a Wiener process, i.e., the independent ...
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Estimator of Bessel function?

I am trying to estimate the parameters of the modified Bessel function of the first kind for integer order case. $I_n(wt) = \sum\limits_{m=0}^\infty \frac{1}{m!(m+n)!}(\frac{wt}{2})^{2m+n}$ In ...
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Is the absolute value of the difference between two Poisson distributions a Poisson distribution?

What is the distribution of the absolute value of the Skellam distribution?
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How to compare rates of occurence in consecutive time series count data?

My data consists of occurrences of words in time windows. E.g.: Day; Word; Frequency 1; "dog"; 45 1; "cat"; 2 ... 2; "dog"; 90 2; "cat"; 4 ... I would like to ...
dimitris fekas's user avatar
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The difference between discrete and continuous variables

Is the number of hydrogen bonds or the number of rings in a molecule a discrete or a continuous variable ? Can I say that the number of rings: 1, 3, 4, $\dots$ is a continuous variable because in ...
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Poisson processes

I have two realizations of a poisson stochastic process, they are over the same space with rate $\lambda_{1}$ and $\lambda_{2}$. What is the probability that N elements in both sequences are the same, ...
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How to calculate cumulative Poisson probabilities without adding each one if no. of outcomes is large and hence, adding (without excel) is difficult?

One can calculate the probability of a correct score of a football match by Poisson(Number of Team1 goals, Mean average Team1 goals) x Poisson(Number of Team2 goals, Mean average Team2 goals) So for ...
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What is the distribution of the sample variance of the Skellam distribution?

I want to estimate the parameter $\mu$ using the difference between two Poisson distributions with the same parameter, i.e. a Skellam distribution with $\mu_1=\mu_2 = \mu$. I can calculate the ...
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Distribution that describes the difference between negative binomial distributed variables?

A Skellam Distribution describes the difference between two variables that have Poisson distributions. Is there a similar distribution that describes the difference between variables that follow ...
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