Discussion
Let X follow a Poisson distribution with probability mass function P(x = 1) = P(x = 3). What is the mean value of this distribution ?
Answer: C
The mean is \(\sqrt{6}\), about 2.45.
For a Poisson distribution with mean \(\lambda\), \(P(x) = e^{-\lambda}\lambda^{x}/x!\). Setting the probabilities of 1 and 3 equal:
\[ \begin{aligned} \frac{e^{-\lambda}\lambda^{1}}{1!} &= \frac{e^{-\lambda}\lambda^{3}}{3!} \\ \lambda &= \frac{\lambda^{3}}{6} \\ \lambda^{2} &= 6,\quad \lambda = \sqrt{6} \end{aligned} \]
The mean must be positive, so the negative root is rejected. \(\sqrt{2}\) and \(\sqrt{3}\) come from using 2 or 3 instead of 3! = 6 in the denominator. 7 has no basis in the working. Since a Poisson mean equals its variance, the variance is also \(\sqrt{6}\).
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