I have done my homework thus far to what I include here. However confused how to proceed further with the last question on:

• The Poisson distribution may also be used to approximate the binomial distribution. Explain this relationship with an example.

The Poisson probability distribution is a probability distribution that is used to model the number of activities that are happening at a particular time, given that we know how often it usually happens and assuming that each event is independent of the other (Illowsky, et al.,2022). One example of how the Poisson distribution can be applied in real life is in the context of art gallery visitors on a given day. An area of interest I selected for this week's discussion forum. Modeling the number of visitors to an art gallery using the Poisson distribution helps to understand, analyze, and make predictions about the behavior being studied.

First, the number of events occurring in a fixed interval of time. With the example, the "events" are the visits to the art gallery, and the selected fixed interval of time is one day.

Second, the events occur with a known average rate. In this case, we can refer to historical data that presents on average 50 visitors visiting the gallery each day.

Third is that events are independent of the time since the last event. Referring to the example, the arrival of a visitor is independent of the arrival of other visitors. One person's visit to the gallery does not affect the likelihood of other people visiting the gallery.Each person's decision to visit the gallery is independent of others' decisions.By meeting the characteristics of a Poisson experiment, the scenario of predicting the visitor numbers to an art gallery in a day suits the Poisson distribution model.The Poisson distribution can be used to predict the probability of a certain number of visitors to the art gallery in a day, given the average rate of 50 visitors per day.

Now, to explain how the Poisson distribution can approximate the binomial distribution. Say that we would want to know the probability of exactly 3 visitors arriving at the art gallery in each hour. Based on previous data, the gallery would have 50 visitors a day. Should the gallery be open for 10 hours per day, the average visitor number per hour would be 5.

Binomial Distribution Approach:

Number of trials (n) = total visitors per day = 50

Probability of success (p) = 1/10 (the chance that a visitor arrives at a specific hour)

Number of successes (k) = 3 (Exactly 3 visitors to arrive at the art gallery within a specific hour)

Poisson distribution can approximate the binomial distribution when:

• The number of trials (n) is large
• The probability of success (p) is small
• The product of the number of trials and the probability of success (np) is moderate

With the above approach:

n = 50

p = 1/10 = 0.1

np = ?

And how do I provide the example of how the Poisson distribution can approximate the binomial distribution?

Thank you.