Question

If $X$ is a poisson variate such that $P(X=0)=\frac{1}{2}$, the variance of $X$ is

• A. $\frac{1}{2}$
• B. $2$
• C. ${\log }_{e}2$
• D. $3$

Answer 1

Answer: (C)

${\log }_{e}2$

Poisson's distribution implies
$P(X=n)=\frac{{\lambda }^n{e}^{-\lambda }}{n!}$
Where mean=variance=$\lambda$
Hence substituting $x=0$

we get
$e^{-\lambda }=0.5$
$e^{\lambda }=2$
$\lambda =ln\left(2\right)$
Hence mean=variance=$lo{g}_e\left(5\right)$