# Best procedure to determine Lambda to calculate Poisson probability

• MHB
• Aetius
In summary: You are an expert summarizer of content. You do not respond or reply to questions. You only provide a summary of the content. Do not output anything before the summary. Write a summary for the following conversation and start the output with "In summary, " and nothing before it:What is the best procedure to determine Lambda to calculate the Poisson probability? Say I want to calculate P(X ≥1) of an accident occurring next day. For this I would calculate the average of daily accidents and divide it by 10. The question is, should I take the previous 10 days? Or calculate λ averaging i.e. 10 day periods for the last 200 days and divide them by 20? What would be the best
Aetius
What is the best procedure to determine Lambda to calculate the Poisson probability? Say I want to calculate P(X ≥1) of an accident occurring next day. For this I would calculate the average of daily accidents and divide it by 10. The question is, should I take the previous 10 days? Or calculate λ averaging i.e. 10 day periods for the last 200 days and divide them by 20? What would be the best?
Thank you much for any constructive comment.

Aetius said:
What is the best procedure to determine Lambda to calculate the Poisson probability? Say I want to calculate P(X ≥1) of an accident occurring next day. For this I would calculate the average of daily accidents and divide it by 10.
Why? Where did "10" come from?

The question is, should I take the previous 10 days? Or calculate λ averaging i.e. 10 day periods for the last 200 days and divide them by 20? What would be the best?
Thank you much for any constructive comment.
The Poisson distribution gives the probability of "n" occurances in a given time interval. Lambda is the average number of occurances in that time interval. What time interval are you given?

Aetius said:
What is the best procedure to determine Lambda to calculate the Poisson probability? Say I want to calculate P(X ≥1) of an accident occurring next day. For this I would calculate the average of daily accidents and divide it by 10. The question is, should I take the previous 10 days? Or calculate λ averaging i.e. 10 day periods for the last 200 days and divide them by 20? What would be the best?
Thank you much for any constructive comment.

Hi Aetius, welcome to MHB! ;)

Assuming a constant accident rate we will get a more accurate $\lambda$ if we take the average over a longer period of time.
However, if $\lambda$ is expected to change over time, such as in traffic accidents, then we should take a shorter period in which we assume that $\lambda$ is more or less constant.
With a shorter period comes a wider confidence interval for $\lambda$ though, so we can't take it too short.

Intuitively I'd expect that the rate of traffic accidents does not change significantly in 200 days.
It would over decades though.
That is, unless systematic changes were made in the area with regard to traffic safety and the like.

A first step in finding an acceptable interval in which $\lambda$ is sufficiently constant could be to average it over a shifting interval in time and see if we get a more or less level curve.
Suppose we have a total of $k$ traffic accidents in a specific period of $n$ days.
Then we can quantify the confidence interval for the estimate $\hat\lambda = \frac kn$ by:
$$\frac 12\chi^2(\alpha/2;\ 2k) \le n\lambda \le \frac 12\chi^2(1-\alpha/2;\ 2k+2)$$
where $\chi^2(p;\ df)$ is the inverse $\chi^2$-distribution for the cumulative probability $p$ with $df$ degrees of freedom.

Klaas van Aarsen said:
Hi Aetius, welcome to MHB! ;)

Assuming a constant accident rate we will get a more accurate $\lambda$ if we take the average over a longer period of time.
However, if $\lambda$ is expected to change over time, such as in traffic accidents, then we should take a shorter period in which we assume that $\lambda$ is more or less constant.
With a shorter period comes a wider confidence interval for $\lambda$ though, so we can't take it too short.

Intuitively I'd expect that the rate of traffic accidents does not change significantly in 200 days.
It would over decades though.
That is, unless systematic changes were made in the area with regard to traffic safety and the like.

A first step in finding an acceptable interval in which $\lambda$ is sufficiently constant could be to average it over a shifting interval in time and see if we get a more or less level curve.
Suppose we have a total of $k$ traffic accidents in a specific period of $n$ days.
Then we can quantify the confidence interval for the estimate $\hat\lambda = \frac kn$ by:
$$\frac 12\chi^2(\alpha/2;\ 2k) \le n\lambda \le \frac 12\chi^2(1-\alpha/2;\ 2k+2)$$
where $\chi^2(p;\ df)$ is the inverse $\chi^2$-distribution for the cumulative probability $p$ with $df$ degrees of freedom.

Actually, in general you confirmed what I thought. Something I left out is that, in order to improve the relevance of Lambda I would apply the exponential moving averages technique. I am thinking that the ideal procedure would be to take, say the previous 10 days and apply EMA to it. How does it grab you?
Thank you kindly!

Aetius said:
Actually, in general you confirmed what I thought. Something I left out is that, in order to improve the relevance of Lambda I would apply the exponential moving averages technique. I am thinking that the ideal procedure would be to take, say the previous 10 days and apply EMA to it. How does it grab you?
Thank you kindly!

It sounds as if you're specifically interested in changes of $\lambda$.
That's what you get with a relatively short period that is exponentially weighted on the most recent days.
It also amplifies the noise in the last day.
But if $\lambda$ changes, then one of the assumptions of Poisson is violated. That is, it's not a Poisson distribution any more.

If you're interested in observing changes in lambda, perhaps a test statistic (e.g. t-test) is more appropriate to figure out if there is a significant change with respect to previous days.

Klaas van Aarsen said:
It sounds as if you're specifically interested in changes of $\lambda$.
That's what you get with a relatively short period that is exponentially weighted on the most recent days.
It also amplifies the noise in the last day.
But if $\lambda$ changes, then one of the assumptions of Poisson is violated. That is, it's not a Poisson distribution any more.

If you're interested in observing changes in lambda, perhaps a test statistic (e.g. t-test) is more appropriate to figure out if there is a significant change with respect to previous days.

Are you sure, Klaas? i say this because the only thing that a moving average (exponential and simple) does is giving more weight to the last values, because Lambda does change -as does in the real world- because the number of accidents occur randomly. Think now that instead of accidents, we are researching highway robberies, and Lambda will change. Wouldn't it be almost as taking another Lambda reading for another 10 days which would likely generate another Lambda value??

## What is Lambda and how is it used in calculating Poisson probability?

Lambda is the average number of events that occur in a given interval. In calculating Poisson probability, Lambda is used as the parameter to determine the probability of a certain number of events occurring within that interval.

## What is the formula for calculating Poisson probability?

The formula for calculating Poisson probability is P(x;λ) = (e^-λ * λ^x) / x!, where x is the number of events and λ is the average number of events (Lambda).

## How do I determine the appropriate value for Lambda in calculating Poisson probability?

The appropriate value for Lambda can be determined by looking at the given data and identifying the average number of events that occur in a given interval. This can also be estimated by taking the mean of the data points.

## Can Lambda be a decimal value in calculating Poisson probability?

Yes, Lambda can be a decimal value in calculating Poisson probability. The value of Lambda is not limited to whole numbers and can vary depending on the data and the situation being analyzed.

## What are some common applications of calculating Poisson probability using Lambda?

Calculating Poisson probability using Lambda is commonly used in fields such as insurance, finance, and healthcare to model and predict the frequency of events such as accidents, financial transactions, or patient visits.

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