Allied & ICAR Exam Bank • ALLIED_1005
MCQ • 1 Mark
Section 1: Engineering Mathematics
Probability and Statistics
Subtopic: Probability distributions
Question Statement:
For a Poisson distributed random variable $X$, if $P(X = 1) = 2 P(X = 2)$, then the variance of $X$ is:
Options:
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(A) 0.5
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(B) 1.0
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(C) 2.0
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(D) 4.0
Step-by-Step Solution & Explanation
$P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$. Given $e^{-\lambda} \lambda = 2 \frac{e^{-\lambda} \lambda^2}{2} = e^{-\lambda} \lambda^2$. Thus $\lambda = \lambda^2 \implies \lambda = 1$ (since $\lambda > 0$). In a Poisson distribution, $\text{Var}(X) = \lambda = 1.0$.
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