To calculate the Poisson probability $P(X = 5)$ with $\lambda = 9$, we'll use the same Poisson probability formula:

$P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}$

Here:

• $\lambda = 9$
• $k = 5$
• $e \approx 2.71828$

Let's calculate the probability.The Poisson probability $P(X = 5)$ for $\lambda = 9$ is approximately 0.0607.

Do you need any more details or have further questions?

Here are five related questions to think about:

1. How would $P(X = 5)$ change if $\lambda$ were increased to 10?
2. What is the cumulative probability $P(X \leq 5)$ for $\lambda = 9$?
3. How do you use the Poisson distribution in queuing theory?
4. How does the skewness of a Poisson distribution change as $\lambda$ increases?
5. What conditions must be met for a Poisson distribution to be a good approximation of a binomial distribution?

Tip: In a Poisson distribution, if $\lambda$ is large, the distribution becomes approximately normal.