Simplifying the Poisson Distribution Formula: Integration and Infinite Series

In summary, the Poisson distribution formula is a mathematical equation used to calculate the probability of a certain number of events occurring within a specific time interval. It is used to model the probability of rare events and has assumptions of a constant average rate of occurrence, independence between events, and a low probability of multiple events occurring simultaneously. The formula is calculated by plugging in the desired number of events, the average rate of occurrence, and the factorial of the number of events. It is also used as an approximation for the normal distribution when the average rate of occurrence is large.
  • #1
ak123456
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b]1. Homework Statement [/b]
prove that

[tex]\sum[/tex]( (e^(-u)) (u(^(x)) )/x! (from x=o to n ) = [tex]\int[/tex] ( (e^(-y))(y^n) )dy/n! (from u to infinite )


Homework Equations





The Attempt at a Solution


i know that the left is Poisson distribution formula but how to do with the 'sum' ?
and the right one i got a infinite series , use integration by part .but i don not know how to simplicity it .
is there anything else i can use for this question ?
 
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any help?
 

Related to Simplifying the Poisson Distribution Formula: Integration and Infinite Series

What is the Poisson distribution formula?

The Poisson distribution formula is a mathematical equation used to calculate the probability of a certain number of events occurring within a specific time interval, given the average rate of occurrence.

When is the Poisson distribution formula used?

The Poisson distribution formula is used to model the probability of rare events, such as the number of customers entering a store in a given hour or the number of earthquakes in a year.

What are the assumptions of the Poisson distribution formula?

The assumptions of the Poisson distribution formula include a constant average rate of occurrence, independence between events, and a low probability of multiple events occurring simultaneously.

How is the Poisson distribution formula calculated?

The Poisson distribution formula is calculated by plugging in the desired number of events (k), the average rate of occurrence (λ), and the factorial of k into the formula: P(k;λ) = (e^-λ * λ^k) / k!

What is the relationship between the Poisson distribution formula and the normal distribution?

The Poisson distribution formula is often used as an approximation for the normal distribution when the average rate of occurrence is large. This is known as the Poisson approximation to the normal distribution.

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