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:

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

Let me calculate this for you.It seems like I can’t do more advanced data analysis right now. Please try again later. Would you like me to explain the steps to solve it manually or provide further clarification on Poisson distribution?

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.