20
votes
Accepted
Is a Poisson minus a constant still a Poisson?
You should incorporate your estimate of the shift into your analysis
As others have pointed out, no, this is not a Poisson distribution (it is actually a shifted Poisson distribution). The bigger ...
- 133k
19
votes
Is a Poisson minus a constant still a Poisson?
Is a Poisson minus a constant still a Poisson?
No. To begin with, the support of a Poisson is the non-negative integers.
But you didn't subtract a constant from a Poisson, you subtracted a constant ...
- 290k
15
votes
Is a Poisson minus a constant still a Poisson?
You know that with a Poisson, the mean equals the variance. Adding a constant to a random variable does not alter the variance but changes the mean. Therefore, a Poisson minus a constant cannot still ...
- 38.3k
14
votes
Please explain the waiting paradox
More on buses... Sorry to butt into the conversation so late in the discussion, but I have been looking at Poisson processes lately... So before it slips out of my mind, here is a pictorial ...
- 26.9k
14
votes
Distribution of half-life from radioactive decay
You ask about the distribution of an order statistic $X_{(k)}$ of $N$ independent and identically distributed random variables $X_1, X_2, \ldots, X_N$ and $k=N/2$ when $N$ is even and $k=(N+1)/2$ when ...
- 334k
13
votes
Accepted
Help me understand poisson.test?
This is an R function that implements a hypothesis test for differences in means. It is analogous to the ?t.test function, except where that assumes the data are normally distributed (in the ...
13
votes
Accepted
MLE for a homogeneous Poisson process?
You can use Maximum Likelihood Estimation, either with synchronous data (time-binned data) or asynchronous data (time-stamped data). The likelihood function changes accordingly.
For time-binned (or ...
- 518
11
votes
Gamma Conjugate Prior & Poisson Process
Your prior is $\lambda\sim\mathcal G(a, b)$, i.e.
$$\pi(\lambda)\propto \lambda^{a-1}e^{-b\lambda}.$$
To get the posterior, multiply the prior by the likelihood:
$$\begin{eqnarray}
\pi\left(\lambda|...
- 2,116
9
votes
Accepted
Compound Poisson random variable
We'll denote $\mu_N = \text{E}(N)$, $\mu_X = \text{E}(X)$ and $\sigma_N^2 = \text{Var}(N)$. To find the covariance we can use the formula
\begin{align}
\text{Cov}(S, N) &= \text{E}(SN) - \text{E}...
- 12.2k
7
votes
Accepted
Radius of circle to ensure certain probability of encountering at least 1 animal
Because you are concerned about finding small numbers of subjects (one), and you suppose (in a comment) that the subject locations are independent, it is reasonable to model the occurrences of animals ...
- 334k
6
votes
Poisson process: getting a poisson from an exponential assumption
I'll leave it to you to fill the details of this:
The exponential waiting time implies that an infinitesimal interval of length $dt$ will have probability $\lambda dt$ of seeing a jump. Since the ...
- 14.1k
6
votes
Accepted
Does exponential waiting time for an event imply that the event is Poisson-process?
Not necessarily a Poisson process
The answer is already sort of given by WHuber in the comments. You need more (restricting) assumptions before the exponential waiting time is to be considered a ...
- 86.5k
6
votes
Alternating between two states {A, B} each with exp distributed durations. What's the probability of state=A at time t?
As @LuisCiadella Comments, this is a Markov Chain. The Markovian structure is inherited from the 'memoryless' property of exponential distributions. Because of its simple structure, it is easy to give
...
- 57.6k
6
votes
Accepted
Finding the probability of survival of an insurance company
Here is my take. Using vectorization can make this a little easier. Comments indicate my logic
...
- 39.6k
6
votes
Accepted
Can Negative Binomial parameters be treated like Poisson?
I put up some code to perform this task in PyMC3, since you mentioned it in the question. The first part, which you seem to already be familiar with, would be fitting the model to get a posterior ...
- 2,690
6
votes
Accepted
How to calculate the confidence interval of a mean on web search findability
Here are examples, using R, of the three kinds of confidence intervals
mentioned in my comment (a), based on 30 observations from $\mathsf{Pois}(\lambda=50).$ [The nonparametric bootstrap CI uses the ...
- 57.6k
6
votes
Are these two equivalent forms for the likelihood of a Poisson point process?
I would consider only the first one as being the likelihood of a Poisson point process. It is unclear what the second expression is aiming at.
Point processes are complex and so is the formulation of ...
- 1,514
5
votes
Accepted
How to simulate Poisson arrival times if the rate varies with time?
There are already several reasonable answers, but to try and add some clarity, here I will try to synthesize these, and do some verification on the resulting algorithm.
Homogeneous Process
For a ...
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5
votes
Accepted
What is mean by the term "constant rate" in Poisson distribution?
I think your question is about hazard rates. The connection between Poisson and Exponential is also addressed here in detail: Relationship between poisson and exponential distribution
The typical ...
- 1,076
5
votes
Accepted
Conditional distribution of arrival times in Poisson process
An interesting property of Poisson processes is that each event can be considered as "placed" independently and uniformly at a given time $t$ in $[0,T]$ (just like rain drops falling uniformly over ...
- 1,356
5
votes
Conditional distribution of arrival times in Poisson process
Since the inter-arrival times are independent exponentially distributed, the joint pdf of the $n$ first arrival times is
\begin{align}
f(t_1,\dots,t_n)
&=f(t_1)f(t_2|t_1)\dots f(t_n|t_{n-1})
\\&...
- 11.7k
5
votes
Poisson regression for count data that is not Poisson distributed?
There are alternatives to Poisson regression, which are typically more general. The Poisson distribution, after all, has only a single parameter, and is equidispersed, which is often a very ...
- 131k
4
votes
Accepted
Distribution of inter arrival times in a Poisson process
The discussion in your book is not phrased correctly in some aspects, but first let me address your question about conditioning on an event of probability $0$; something that is explicitly forbidden ...
- 47.8k
4
votes
How to prove the independent and stationary increment of a poisson process?
This is modified from my answer to a related question. The only thing this answer assumes you already know about the Poisson process is that $\mathbb{P}(N_t=n)$ is a Poisson-distributed random ...
- 6,479
4
votes
Fit to a non-homogeneous Poisson process
Assume you want to model the intensity $\lambda$ by some $\lambda_\theta$ where $\theta$ is a parameter. For a temporal Poisson point process on the interval $[0,T]$ the log-likelihood is known and ...
- 1,514
4
votes
How to simulate Poisson arrival times if the rate varies with time?
There are several methods. For simple rate functions like yours I find the inversion approach to be the easiest. See algorithm 3 of this paper; it's 5 lines. Edit: and the algorithm is:
...
- 806
4
votes
Accepted
Motivation for gamma distribution with a non-integer parameter
It seems you are asking for "real-life" examples where gamma distributions are used to model some real-world observables represented by random variables. There are many such examples. Take the Erlang ...
- 82.8k
4
votes
Equation for Inverse Poisson CDF
Quantile function: If you already have the CDF $F$ available, you can write the quantile function for the Poisson distributon as:
$$\begin{align}
Q(p)
&\equiv \inf \Big\{ x = 0,1,2,... \Big| \ p \...
- 133k
4
votes
Accepted
Histogram for a compound poisson process
You wouldn't draw a histogram of a Poisson process at all. It's an increasing sequence of values (0,1,2,...) across a set of continuous times:
You could draw something like a histogram of ...
- 290k
4
votes
Accepted
Normal approximation on (what it looks like) a poisson
The probability of "success" in a single draw is $p=1/6$, since there are 6 possibilities out 36 to have a sum of 7. If the number of draws were constant let's say $N=180$, then this is a binomial ...
- 306
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