Poisson distribution and random processes

[paul-g]

Hello!

I am writing because I recently became interested in probability distributions, and I have to you a few questions.

Poisson distribution is given as a probability:

$f(k;\lambda)=\frac{\lambda^{k}e^{-\lambda}}{k!}$

But what is lambda?

Suppose that we consider as an unrelated incident falling raindrops. If these drops fall 100 in 1 on a surface second how much $\lambda$ will be?

How to check if the falling drops of rain or some other unrelated events are described in this distribution?

 

[cdotter]

From Wikipedia: λ is a positive real number, equal to the expected number of occurrences during the given interval. For instance, if the events occur on average 4 times per minute, and one is interested in the probability of an event occurring k times in a 10 minute interval, one would use a Poisson distribution as the model with λ = 10×4 = 40.

I'm not sure what you mean by "100 in 1."

 

[paul-g]

Meant falling about 100 drops per second on the area.

Im curious how can be checked whether a process is described in this distribution.

For example, the number of drops falling on the glass. How can I check if they are described Poisson distribution.

 

[cdotter]

I think you would also need a time, since λ measures the expected number of occurrences and not the rate.

I've only taken one statistics course, but generally the problem will explicitly tell you if it follows a Poisson distribution. There are tests for Poission distributions if you're doing "real world" statistics, but I don't know anything about such tests. Maybe try googling "Poisson distribution test."

 

[Matrix0]

Hello!

First of all, let me explain what Poisson distribution is and its relationship to random processes. Poisson distribution is a discrete probability distribution that describes the probability of a certain number of events occurring within a fixed time or space interval, given the average rate of occurrence of those events. It is often used to model random processes, where events occur independently and at a constant rate.

Now, to answer your question about lambda, it is a parameter in the Poisson distribution that represents the average rate of occurrence of the events. In other words, it is the expected number of events that will occur in the given time or space interval.

In the case of falling raindrops, if we assume that the drops fall at a constant rate of 100 per second, then the value of lambda would be 100. This means that on average, we would expect 100 raindrops to fall in 1 second.

To check if the falling drops of rain or any other unrelated events can be described by the Poisson distribution, we can look at certain characteristics of the data. For example, if the events occur independently and at a constant rate, then the Poisson distribution would be a good fit. We can also use statistical tests to compare the observed data to the expected distribution and determine if they are a good match.

I hope this helps answer your questions about Poisson distribution and random processes. If you have any further questions, please don't hesitate to ask. Keep exploring and learning about probability distributions, they are fascinating tools in understanding and predicting various phenomena in the world around us.