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The number $X$ of people in a certain town who get the influenza follows a Poisson distribution. The proportion of people who did not get the flu is $0.01$. Find the probability mass function of $X$.

Ok so the probability mass function formula is $e^{-\lambda}\frac{\lambda^k}{k!}$. I saw the solution but I do not understand how this was solved?

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  • $\begingroup$ This does not make sense. If the population of the town is $N$, then $0.99 N$ is the number of people who got the flu, i.e. it is an observation of $X$. You could use that as an estimator of the parameter $\lambda$, but you can't say what the true $\lambda$ is. $\endgroup$ – Robert Israel Jun 22 '17 at 17:59
  • $\begingroup$ Of course, the number who get the flu can't be more than the population of the town, while a Poisson random variable can take arbitrarily large values, so it can't really be Poisson: Poisson is just an approximation. $\endgroup$ – Robert Israel Jun 22 '17 at 18:07
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The only meaningful exercise is

The number $X$ of people in a certain town who get the influenza follows a Poisson distribution. The probbility that no one did get the flu is $0.01$. Find the probability mass function of $X$.

Thus the equation is $P(X=0)=e^{-\lambda}\cdot \frac{\lambda^0}{0!}=0.01$

Thus $\lambda=\ln(100)$

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