Simplifying the Poisson Distribution Formula: Integration and Infinite Series

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SUMMARY

The forum discussion centers on proving the equivalence of the Poisson distribution formula expressed as a summation and an integral. Specifically, it addresses the equation \(\sum_{x=0}^{n} \frac{e^{-u} u^{x}}{x!} = \frac{1}{n!} \int_{u}^{\infty} e^{-y} y^{n} dy\). Participants discuss the left side as the Poisson distribution and explore integration by parts as a method to simplify the infinite series on the right side. The conversation emphasizes the need for clarity in transitioning from summation to integration.

PREREQUISITES
  • Understanding of Poisson distribution and its properties
  • Knowledge of integration techniques, particularly integration by parts
  • Familiarity with infinite series and their convergence
  • Basic proficiency in calculus, specifically with summation and integration
NEXT STEPS
  • Study the derivation of the Poisson distribution formula in detail
  • Learn advanced integration techniques, focusing on integration by parts
  • Explore convergence criteria for infinite series
  • Investigate applications of the Poisson distribution in real-world scenarios
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Students and educators in mathematics, particularly those studying probability theory and calculus, as well as anyone looking to deepen their understanding of the Poisson distribution and its applications in statistics.

ak123456
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b]1. Homework Statement [/b]
prove that

\sum( (e^(-u)) (u(^(x)) )/x! (from x=o to n ) = \int ( (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?
 

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