Question

If 6 is the mean of a Poisson distribution, then $P(X \geq 3) =$

• A. $1 - \dfrac{25}{e^6}$
• B. $\dfrac{e^{-6}}{25}$
• C. $\dfrac{24 - 25}{e^6}$
• D. $e^3$

Hint:

Use complement rule for cumulative Poisson probabilities when $P(X \geq k)$ is asked.

Answer 1

The Correct Option is A

Given $\lambda = 6$, and $X \sim P(6)$
We use complement: $P(X \geq 3) = 1 - P(X<3) = 1 - [P(0) + P(1) + P(2)]$
$P(0) = \dfrac{e^{-6} \cdot 6^0}{0!} = \dfrac{1}{e^6}$
$P(1) = \dfrac{e^{-6} \cdot 6^1}{1!} = \dfrac{6}{e^6}$
$P(2) = \dfrac{e^{-6} \cdot 6^2}{2!} = \dfrac{36}{2e^6} = \dfrac{18}{e^6}$
Sum = $\dfrac{1 + 6 + 18}{e^6} = \dfrac{25}{e^6}$
Therefore, $P(X \geq 3) = 1 - \dfrac{25}{e^6}$