In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.
The mean number of customers who arrive at the checkout counters each minute is 4. Create a Poisson distribution with mu = 4 for x = 0 to 20. Compare your results with the histogram shown at the upper right.

Verified step by step guidance

1. Understand the problem: The Poisson distribution is used to model the number of events (in this case, customer arrivals) occurring in a fixed interval of time. The mean (μ) is given as 4, and we are tasked with creating a Poisson distribution for x = 0 to 20.
2. Recall the formula for the Poisson probability mass function (PMF): P(X = x) = (e^(-μ) * μ^x) / x!, where μ is the mean, x is the number of events, and e is the base of the natural logarithm (approximately 2.718).
3. For each value of x from 0 to 20, calculate the probability using the Poisson PMF formula. For example, for x = 0, P(X = 0) = (e^(-4) * 4^0) / 0!. Repeat this calculation for x = 1, x = 2, ..., up to x = 20.
4. Once the probabilities are calculated, create a table or list of the values of x (from 0 to 20) and their corresponding probabilities. This will represent the Poisson distribution for the given mean μ = 4.
5. To compare your results with the histogram, plot the calculated probabilities as a bar graph (histogram) with x on the horizontal axis and P(X = x) on the vertical axis. Visually compare the shape and distribution of your graph with the provided histogram.

Key Concepts

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. It is characterized by the parameter 'mu' (λ), which represents the average number of events in the interval. In this case, with mu = 4, it models the number of customers arriving at the checkout counters per minute.

Mean (Expected Value)

The mean, or expected value, of a probability distribution is the long-term average value of repetitions of the experiment it represents. For the Poisson distribution, the mean is equal to the parameter mu (λ). In this scenario, a mean of 4 indicates that, on average, 4 customers arrive at the checkout counters each minute.

Histogram

A histogram is a graphical representation of the distribution of numerical data, where the data is divided into bins or intervals. Each bin's height represents the frequency of data points within that interval. In this context, comparing the histogram of the Poisson distribution with the calculated probabilities helps visualize how the number of customers arriving aligns with the expected distribution.