Consider a random variable to which a Poisson distribution is best fitted. It happens that \({P(x=1)} = \frac{2}{3}{P({x\;=\;2})}\) on this distribution plot. The variance of this distribution will be

Concept:

It follows a Poisson distribution as the probability of occurrence is very small.

\({\rm{Probability}},{\rm{\;P\;}}\left( {{\rm{x\;}} = {\rm{\;r}}} \right) = \frac{{{e^{ - λ }} \;{λ ^r}}}{{r!}}\)

where mean E(x) = variance Var (x) = λ and standard deviation (σ) = \(\sqrt{λ}\)

Calculation:

Given: 

\({P_{x\;=\;1}} = \frac{2}{3}{P_{x\;=\;2}}\)

For Poisson distribution:

\({\rm{Probability}},{\rm{\;P\;}}\left( {{\rm{x\;}} = {\rm{\;r}}} \right) = \frac{{{e^{ - λ }} \;{λ ^r}}}{{r!}}\)

\( \Rightarrow \frac{{{e^{ - λ}}{λ^1}\;}}{{1!}} = \frac{2}{3}\frac{{{e^{ - λ}}{λ^2}}}{{2!}}\)

⇒ λ = 3

For Poisson distribution,

Mean = Variance = λ