Question

If \(X\) is a Poisson random variable with mean \(3\), then \[ P(|X-3|\lt 2) = \] is equal to:

• A. \(\dfrac{9}{2e^3}\)
• B. \(\dfrac{99}{8e^3}\)
• C. \(\dfrac{3}{2e^3}\)
• D. \(\dfrac{1}{3e^3}\)

Hint:

For a Poisson random variable with parameter \(\lambda\), \[ P(X=x)=e^{-\lambda}\frac{\lambda^x}{x!}. \] Always convert inequalities involving \(X\) into the corresponding integer values before computing probabilities.

Answer 1

The Correct Option is B

Step 1: Identify the values of X satisfying \(|X-3|<2\).
\(|X-3|<2 \Rightarrow 1<X<5 \Rightarrow X\in\{2,3,4\}\).

Step 2: Compute using Poisson formula with \(\lambda=3\).
\[P(X=2)=e^{-3}\frac{9}{2},\quad P(X=3)=e^{-3}\frac{27}{6}=e^{-3}\frac{9}{2},\quad P(X=4)=e^{-3}\frac{81}{24}=e^{-3}\frac{27}{8}.\] \[P = e^{-3}\left(\frac{9}{2}+\frac{9}{2}+\frac{27}{8}\right) = e^{-3}\cdot\frac{36+36+27}{8} = \frac{99}{8e^3}.\]
\[\boxed{\frac{99}{8e^3}}\]