POISSON DISTRIBUTION: A startup receives message, on average, 7 text messages

in a 3-hour period timeframe? (a) What is the

probability that the business will receive exactly 9 text

messages in a 3-hour period? (b) What is the probability

that the startup will receive exactly 24 text messages in

8 hours?

The **Poisson distribution** is a discrete probability distribution that describes the probability of a given number of events occurring in a fixed interval of time or space, provided the events occur with a known constant mean rate and independently of the time since the last event.

The **Poisson probability mass function (PMF)** is given by:

\[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
\]

Where:
- \( X \) is the random variable (number of events),
- \( \lambda \) is the expected number of events in a given time period,
- \( k \) is the actual number of events,
- \( e \) is Euler's number (approximately 2.71828).

### (a) What is the probability that the startup will receive exactly 9 text messages in a 3-hour period?

In this case:
- The average rate of messages per 3-hour period, \( \lambda \), is 7.
- We want to find the probability of receiving exactly 9 text messages, so \( k = 9 \).

Using the formula:

\[
P(X = 9) = \frac{7^9 e^{-7}}{9!}
\]

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Here are five related questions you might find useful:
1. What are some common applications of the Poisson distribution in business settings?
2. How does the Poisson distribution differ from the binomial distribution?
3. What is the probability of receiving more than 10 messages in a 3-hour period?
4. How would you compute the cumulative probability for receiving fewer than 5 messages?
5. How does changing the time interval affect the Poisson parameter \( \lambda \)?

**Tip:** The Poisson distribution assumes events happen independently and at a constant average rate, which is key to accurate modeling.

POISSON DISTRIBUTION: A startup receives, on average, 7 text messages in a 3-hour period timeframe. (a) What is the probability that the business will receive exactly 9 text messages in a 3-hour period? (b) What is the probability that the startup will receive exactly 24 text messages in 8 hours?

Explore how to calculate the probability of a startup receiving exactly 9 messages in 3 hours and 24 messages in 8 hours using the Poisson distribution. Understand the core concepts of probability, including the Poisson distribution formula and the role of the average rate of occurrence (λ). Suitable for advanced high school or college-level students.

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