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If $X$ is a poisson variate such that $P(X=1)$ $=P(X=2)$, then $P(X=4)$ is equal to
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1785 Upvotes
Verified Answer
The correct answer is:
$\frac{2}{3 \mathrm{e}^{2}}$
Given : $\mathrm{P}(\mathrm{X}=1)=\mathrm{P}(\mathrm{X}=2)$
$$
\begin{array}{l}
\frac{\mathrm{e}^{-\lambda} \lambda}{1 !}=\frac{\mathrm{e}^{-\lambda} \lambda^{2}}{2 !} \\
\Rightarrow \lambda=2 \\
\therefore \mathrm{P}(\mathrm{X}=4)=\frac{\mathrm{e}^{-2} 2^{4}}{4 !}=\frac{2}{3 \mathrm{e}^{2}}
\end{array}
$$
$$
\begin{array}{l}
\frac{\mathrm{e}^{-\lambda} \lambda}{1 !}=\frac{\mathrm{e}^{-\lambda} \lambda^{2}}{2 !} \\
\Rightarrow \lambda=2 \\
\therefore \mathrm{P}(\mathrm{X}=4)=\frac{\mathrm{e}^{-2} 2^{4}}{4 !}=\frac{2}{3 \mathrm{e}^{2}}
\end{array}
$$
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