12
votes
Accepted
What is the difference between the Poisson distribution and the uniform distribution?
@scortchi has the right answer. To summarize:
The arrival time stamps are uniformly distributed.
The inter-arrival times are exponentially distributed.
The count of arrivals per uniform time period ...
- 331
5
votes
Accepted
If the mean is equal to the standard deviation, what is the general likelihood that the underlying distribution is normal vs exponential?
Any talk here of probability must at best be informal without a precise idea of what set-up you are notionally sampling from. But it's clear that a normal with mean and SD equal must have both ...
- 59.5k
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
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
4
votes
Number of events in a segment if waiting times are drawn from a mixture of two exponential distributions
Part 1: fixed waiting times
What is the probability for $n$ events to occur over a period of time $t$, if the duration of each individual event is $\tau_1$ with the probability $p$ and $\tau_2$ ...
- 86.5k
3
votes
Accepted
For binomial success / fail data, how to calculate if the time since last success is out of the normal range
If the ordinary probability of Pass is $p,$ then you are right that on average you should expect a Pass every
$1/p$ days. So roughly speaking, if $p = 1/10$ you might start to get suspicious something ...
- 57.6k
3
votes
Are arrivals uniformly distributed over a time interval when inter-arrival times are poisson distributed
Will the arrivals be uniformly distributed over $T$?
No -- and yes.
In any one realization of this process the arrival times will be random. They usually will not be uniformly spread throughout the ...
- 334k
3
votes
Accepted
Poisson process and queuing system
If interarrival times ($IA$) are i.i.d. $IA\sim \text{Exponential}(\lambda)$ then the $n$th arrival time is $S_n \sim \text{Erlang}(n,\lambda)$. as a quick check, you can see that $S_1 \sim \text{...
- 1,595
2
votes
Accepted
Finite state machine with gamma distributed waiting times
Note that this is NOT an attempt to fully answer the problem, but to show how to overcome the lack of the Markov property for a special case that may not apply - one that is far too long to put in ...
- 41.1k
2
votes
Why does the superposition of two processes with geometrically distributed interarrival times not result in a process with similar distribution?
A process with "geometrically distributed" inter-arrival times is a discrete-time Bernoulli random process (the random variables $X[n]$ are i.i.d. Bernoulli random variables), and an arrival ...
- 47.8k
2
votes
Accepted
Non-parametric (smoothed) estimate of current rate
This is a common problem in neuroscience, and the same methods can be applied to your problem. I'll describe the neuroscience problem briefly, because this will help to interpret the papers below. The ...
- 33.2k
2
votes
Are arrivals uniformly distributed over a time interval when inter-arrival times are poisson distributed
If there are $N_t \sim \mathsf{Pois}(\lambda t)$ events in interval $(0, t),$ with $t > 0,$
then their interarrival times will be $X \sim \mathsf{Exp}(\lambda).$
Consider the event "no arrivals" in ...
- 57.6k
2
votes
Number of events in a segment if waiting times are drawn from a mixture of two exponential distributions
The distribution of waiting times you describe is a hyperexponential distribution. See the Wikipedia entry for details.
each $Y_i$ is an exponentially distributed random variable with rate ...
- 1,988
2
votes
Example of dataset where the data collected at time-points $g(t_1), g(t_2), \ldots$
Any irregular time series fits your description. These are time series where observations are not taken in equally spaced intervals, but at irregular ones - but the sequence is still taken in time ...
- 131k
1
vote
How to model arrival times in discrete event simulation where arrivals vary with time of day and day of week
The first part of this question is an approximate duplicate of
Nonhomogeneous poisson process simulation
When I wrote the question, I was unaware of the name for the arrival distribution I was trying ...
- 133
1
vote
Poisson Distributions: Have Daily Rate and "peak rate", how to figure out hourly rate?
Both of the "M"s in the M/M/1 refer to the assumption of a homogeneous Poisson process. The latter may be (essentially, up to technicalities) described by the Poisson distribution of the event counts ...
- 231
1
vote
Accepted
Fit inter-arrival time to Poisson Distribution/Exponential
The Poisson distribution is for event counts in a given interval of time (in a Poisson process). It is not useful for modelling inter-event times.
If the events follow a Poisson process, the inter-...
- 291k
1
vote
Accepted
Interarrival times of exponential distribution
It turns out my calculation was correct, but my simulation wasn't (wrong normalization --- ooops!). When $n$ iid events are drawn from an exponential distribution of time constant $\tau$, the pdf of ...
- 378
1
vote
How to benchmark an ACD model?
I'm not surprised at the median performing "best" in terms of the absolute error. If there are truly no dynamics and all observations are iid, then the median will minimize the expected absolute error ...
- 131k
1
vote
Calculate predictability of events over time
The model is new and therefor every new word is "predictable"
I don't find this notion so intuitive. Perhaps better to say that given that you have no knowledge of which words are more likely to ...
- 61
1
vote
Accepted
Algorithms for extracting moving average intervals
When this is done electronically it is usually done with an RC (resistance/capacitance) circuit e.g., see link. The circuit shown in that link is not what you want, rather you want the resistor in ...
- 13.3k
1
vote
Accepted
Inter-arrival time of negative binomial distribution
The underlying process is called the Bernoulli process with parameter
$p$ in which the
inter-arrival time is integer-valued, and has a geometric distribution
with parameter $p$. The number of arrivals ...
- 47.8k
Only top scored, non community-wiki answers of a minimum length are eligible