Calculate each Poisson probability:

**a.** *P*(*X* = 5), *λ* = 9

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.

Learn how to calculate the Poisson probability P(X = 5) for λ = 9 using the Poisson probability formula. Understand the concepts of Poisson distribution and its application in probability theory. Suitable for advanced high school students and learners interested in probability.

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