If $X$ is a poisson variate with $P(X=0)=0.8$, then the variance of $X$ is
If $X$ is a poisson variate with $P(X=0)=0.8$, then the variance of $X$ is
- $\log _e 20$
- $\log _{10} 20$
- $\log _e 5 / 4$
- $0$
Solution
Poisson distribution
$
\begin{aligned}
& P(X)=\frac{e^{-m} m^x}{x !} \\
& \therefore \quad P(X=0)=\frac{e^{-m} 1}{1} \\
& \Rightarrow \quad 0.8=e^{-m} \Rightarrow-m=\log _e 0.8 \\
& \Rightarrow \quad m=\log _e \frac{10}{8}=\log _e \frac{5}{4} \\
&
\end{aligned}
$
As, we know in a poisson distribution variance
$
\text { Variance }=\log _e \frac{5}{4} \quad=m
$
Asked in: AP EAMCET 2004
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