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