Exercise · ZW-POISSON-DISTRIBUTION-009 · ★★★ Advanced
A Poisson variable satisfies \(P(X=1)=P(X=2)\) with \(\lambda>0\). What is \(\lambda\)?
Choose one option.
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- C. \(2\)
Explanation
Equating \(\lambda e^{-\lambda}=\tfrac{\lambda^2}{2}e^{-\lambda}\) and simplifying gives \(1=\tfrac{\lambda}{2}\), so \(\lambda=2\).
Review the concept: Poisson distribution
A Poisson variable \(X\) counts independent events, at a constant average rate, in a fixed interval; its parameter \(\lambda\) is the mean number of events and \(P(X=k)\) the probability of observing \(k\).
\[P(X=k)=\frac{e^{-\lambda}\lambda^{k}}{k!},\quad k=0,1,2,\dots\]
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