Question Bank • QB_EM_GATE_005 NAT • 2 Marks

Section 1: Engineering Mathematics

Probability and Statistics

Subtopic: Poisson, normal and binomial distributions

Question Statement:

A random variable $X$ follows a Poisson distribution such that $P(X = 2) = \frac{2}{3} P(X = 1)$. The mean of $X$ is ________ (round off to 2 decimal places).

Answer Key:

1.33

Step-by-Step Solution & Explanation

For a Poisson distribution with parameter $\lambda$: $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$ Given $P(X = 2) = \frac{2}{3} P(X = 1)$: $\frac{e^{-\lambda} \lambda^2}{2!} = \frac{2}{3} \frac{e^{-\lambda} \lambda^1}{1!} \implies \frac{\lambda^2}{2} = \frac{2\lambda}{3} \implies \lambda = \frac{4}{3} \approx 1.333$ Rounding to 2 decimal places: **1.33**.
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