Subtopic: Poisson, normal and binomial distributions
Question Statement:
A random variable $X$ follows a Poisson distribution such that $P(X = 1) = 2 P(X = 2)$. The mean $\lambda$ of the distribution is ________ (answer in integer).
Answer Key:
1
Step-by-Step Solution & Explanation
For a Poisson distribution with parameter $\lambda$:
$P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$
Given $P(X = 1) = 2 P(X = 2)$:
$\frac{e^{-\lambda} \lambda^1}{1!} = 2 \cdot \frac{e^{-\lambda} \lambda^2}{2!}$
$\lambda = 2 \cdot \frac{\lambda^2}{2} = \lambda^2$
Since $\lambda > 0$, dividing both sides by $\lambda$ yields:
$\lambda = 1$