# How to find equations for confidence limits in Poisson distribution?

• Mixer
In summary, to find the confidence limits for λ in a Poisson distribution, we use the equation P(X ≤ k) = ∑ (k = 0 to n) e-λ λk / k!, where k is the observed number of events, n is the maximum possible value for k, and λ is the mean rate of events. We can then solve for λ by setting P(X ≤ k) = 100(1-a)%, where a is the desired level of confidence, to find the upper and lower confidence limits.
Mixer

## Homework Statement

What kind of equations you'll get when trying to find confidence limits 100(1-a) % for λ in Poisson distribution?

## Homework Equations

Poisson distribution P(X=x) = e λx / x! (x=0,1,2 ...)

## The Attempt at a Solution

I made an equation as follows:

Ʃ (k = from k0 to n) e λk / k! = a/2

to solve the confidence limits. Am I doing correct thing here? I cannot solve λ from that equation + I'm not supposed to have infinite sums in my equations... Any help for me?

Last edited:

Your equation is not quite correct. To find the confidence limits for λ in a Poisson distribution, we can use the following equation:

P(X ≤ k) = ∑ (k = 0 to n) e-λ λk / k!

where k is the observed number of events, n is the maximum possible value for k, and λ is the mean rate of events.

To find the confidence limits, we can solve for λ by setting P(X ≤ k) = 100(1-a)%, where a is the desired level of confidence. This will give us two values for λ, one for the lower confidence limit and one for the upper confidence limit.

For example, if we want to find the 95% confidence limits for λ in a Poisson distribution, we would set P(X ≤ k) = 95% and solve for λ. This would give us two values, one for the lower limit and one for the upper limit, which we can then use to determine the range of values for λ with 95% confidence.

## 1. What is a Poisson distribution?

A Poisson distribution is a probability distribution that is used to model the number of occurrences of a specific event within a fixed interval of time or space. It is characterized by one parameter, λ (lambda), which represents the average number of occurrences within the given interval.

## 2. How do I find the equation for confidence limits in a Poisson distribution?

The equation for confidence limits in a Poisson distribution is:
Lower Limit = λ - z√(λ)
Upper Limit = λ + z√(λ)
where z is the z-score corresponding to the desired confidence level. For example, for a 95% confidence level, z = 1.96.

## 3. Can I use a calculator to find the confidence limits in a Poisson distribution?

Yes, many scientific calculators have functions for calculating confidence limits in a Poisson distribution. You can also use online calculators or statistical software such as Excel or SPSS.

## 4. What is the purpose of finding confidence limits in a Poisson distribution?

The purpose of finding confidence limits in a Poisson distribution is to estimate the range of values within which the true mean or average number of occurrences is likely to fall. This can help in making decisions and drawing conclusions based on the data.

## 5. Are there any assumptions or limitations when using confidence limits in a Poisson distribution?

Yes, there are certain assumptions and limitations when using confidence limits in a Poisson distribution. These include having a sufficiently large sample size, independence of observations, and the events being rare and randomly occurring. Additionally, the confidence limits may not be accurate if the true mean is close to zero or if the data is highly skewed.

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