Mean of a Poisson Distribution From Two Relative Probabilities

Suppose we have good reason to suspect that a particular random variable  
  follows a Poisson distribution, so that  
\[Z \sim Po( \lambda )\]
, where  
  is the relevant parameter, typically a frequency rate. Then we can find  
  given the relative probabilities of observations of  
For a Poisson distribution,  
\[P(X=x)=\frac{\lambda^x e^{-\lambda}}{x!}\]
Suppose that  
\[\frac{\lambda^7 e^{-\lambda}}{7!}=2 \frac{\lambda^8 e^{-\lambda}}{8!} \]
Divide by  
\[e^{- \lambda}\]
  to give  
\[\frac{\lambda^7}{7!}=2 \frac{\lambda^8 }{8!} \]
Multiply,y both sides by  
  and divide by  
. Then  
\[8=2 \lambda \rightarrow \lambda =4\]

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