Question

If $X$ is a random Poisson variate such that $P(X=0)=\frac{1}{e}$, then the variance of the same distribution is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (A)

$1$

Poisson's distribution implies
$P(X=n)=\frac{{\lambda }^n{e}^{-\lambda }}{n!}$
Where mean=variance=$\lambda$
Hence $P(x=0)=\frac{1}{e}$
$\frac{{\lambda }^0{e}^{-\lambda }}{0!}=\frac{1}{e}$
$\lambda =1$
Hence mean=variance=$1$