Question

If, in a Poisson distribution $P(X=0)=k$ then the variance is:

• A. $e^{\lambda }$
• B. $\log \frac{1}{k}$
• C. $\frac{1}{k}$
• D. $\log k$

Answer 1

Answer: (B)

$\log \frac{1}{k}$

Poisson distribution $P\left(X\right)=\frac{e^{-\lambda }{\lambda }^x}{x!}$

$P(X=0)=k\Rightarrow \frac{e^{-\lambda }1}{1}=k$

$\Rightarrow e^{-\lambda }=k$

$\therefore k=e^{-\lambda }$

$\Rightarrow \log e^{-\lambda }=\log k$

$\Rightarrow$ $-\lambda \log e=\log k$

$\Rightarrow$ $-\lambda =\log k$

$Variance=\lambda =-\log k$

$\Rightarrow$ $\lambda =\log \left(\frac{1}{k}\right)$