Question

In a poisson distribution $P(X=0)=P(X=1)=k$, then the value of $k$ is

• A. $1$
• B. $\frac{1}{e}$
• C. $e$
• D. $\sqrt{2}$

Answer 1

Answer: (B)

$\frac{1}{e}$

$P(x;\mu )=\frac{e^{-\mu }{\mu }^x}{x!}$

Given,$P(X=0)=P(X=1)=k$

$\frac{e^{-\mu }{\mu }^0}{0!}=\frac{e^{-\mu }{\mu }^1}{1!}$
$e^{-\mu }=e^{-\mu }\mu$
$e^{-\mu }(1-\mu )=0$
Since $e^{-\mu }\ne 0$ , => $\mu =1$
$P(X=0)=\frac{e^{-\mu }{\mu }^0}{0!}=e^{-\mu }=e^{-1}=\frac{1}{e}$