Question

A store opens at 8 A.M. From 8 until 10 A.M. customers arrive at a Poisson rate of four an hour. Between \(10 \mathrm{~A} . \mathrm{M}\). and \(12 \mathrm{P}\).M. they arrive at a Poisson rate of eight an hour. From \(12 \mathrm{P} . \mathrm{M}\). to \(2 \mathrm{P} \mathrm{M}\). the arrival rate increases steadily from eight per hour at 12 P.M. to ten per hour at 2 P.M.; and from 2 to \(5 \mathrm{P}_{-} \mathrm{M}\). the arrival rate drops steadily from ten per hour at \(2 \mathrm{P.M}\). to four per hour at \(5 \mathrm{P} . \mathrm{M} .\). Determine the probability distribution of the number of customers that enter the store on a given day.

Short answer

The probability distribution of the number of customers that enter the store on a given day is a Poisson distribution with a mean of 63 customers per day: \(P(X = k) = \frac{e^{-63} \cdot 63^k}{k!}\) , for non-negative integer values of k.

Step-by-step solution

Identify the time intervals and their corresponding rates.

The store is open from 8 A.M. to 5 P.M., and we have the following information about customer arrivals: a. 8 A.M. to 10 A.M. - Poisson rate = 4 customers per hour b. 10 A.M. to 12 P.M. - Poisson rate = 8 customers per hour c. 12 P.M. to 2 P.M. - Arrival rate increases from 8 to 10 customers per hour (Linear increment) d. 2 P.M. to 5 P.M. - Arrival rate drops from 10 to 4 customers per hour (Linear decrement)

Calculate the expected number of customers for each interval.

For the first two intervals (a and b), we can directly multiply the Poisson rate by the duration (in hours) to find the expected number of customers: a. Expected number of customers from 8 A.M. to 10 A.M. = 4 customers per hour * 2 hours = 8 customers b. Expected number of customers from 10 A.M. to 12 P.M. = 8 customers per hour * 2 hours = 16 customers For intervals c and d, the arrival rate is changing linearly. We can find the average arrival rate and multiply it with the duration: c. Average arrival rate from 12 P.M. to 2 P.M. = (8 customers per hour + 10 customers per hour) / 2 = 9 customers per hour c. Expected number of customers from 12 P.M. to 2 P.M. = 9 customers per hour * 2 hours = 18 customers d. Average arrival rate from 2 P.M. to 5 P.M. = (10 customers per hour + 4 customers per hour) / 2 = 7 customers per hour d. Expected number of customers from 2 P.M. to 5 P.M. = 7 customers per hour * 3 hours = 21 customers

Sum up the expected values for all intervals.

Now, we sum up the expected number of customers for each interval to find the total expected number of customers per day: Total expected number of customers per day = 8 customers (from 8 A.M. to 10 A.M.) + 16 customers (from 10 A.M. to 12 P.M.) + 18 customers (from 12 P.M. to 2 P.M.) + 21 customers (from 2 P.M. to 5 P.M.) = 63 customers

Determine the probability distribution.

Since the arrival of customers follows a Poisson distribution in each interval, and we have found the expected number of customers during each interval, we can now calculate the probability distribution for the total number of customers per day: Let X be a discrete random variable representing the number of customers per day. We know that the expected value of a Poisson distribution with rate λ is equal to λ. Therefore, the probability distribution of the number of customers that enter the store on a given day is a Poisson distribution with a mean of 63 customers per day: \(P(X = k) = \frac{e^{-63} \cdot 63^k}{k!}\) , for non-negative integer values of k. Thus, we determined the probability distribution of the number of customers that enter the store on a given day.

Key concepts

Probability Distribution

Understanding the concept of a probability distribution is like envisioning a map that shows all the possible outcomes of a random event and how likely each outcome is to occur. In simpler terms, it's a way to describe all the 'what-ifs' and their associated chances. There are several types of probability distributions, each suited to specific types of events and data. The Poisson distribution, for instance, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independently of the time since the last event.

For the store opening scenario, we are dealing with a random number of customers arriving per hour, which aligns perfectly with the characteristics of a Poisson distribution. The Poisson distribution is especially useful for managing unlikely events over time, such as receiving a certain number of emails in an hour or counting the number of meteorites of a certain size hitting the Earth. In our store example, it helps the management team predict customer flow and staffing needs based on historical data. To fully grasp the Poisson distribution, it's important to recognize its key properties: it's discrete, meaning it deals with countable events like customers walking into a store; and it has an equal mean and variance, signifying the average rate (expected number of occurrences) is also the distribution's variance (how spread out the occurrences are).

Expected Value

Think of the expected value of a random variable as the average or mean outcome you'd anticipate after many repetitions of a random event. It represents the long-term average or 'center of mass' of a probability distribution. Crucially, it's not the most probable outcome for just one trial but is extremely reliable over a large number of trials.

In mathematical terms, for a discrete random variable, the expected value is calculated by summing the product of each possible value the variable can take and the probability of that value occurring. It gives us a way to quantify what we 'expect' on average, even when randomness is at play. In the context of the store from our exercise, the expected value helps us predict how many customers might arrive in each time interval based on their arrival rates. By calculating the expected number of customers arriving during specific time frames, the store management can optimize resources effectively to manage the customer flow, ensuring the store is neither overstaffed during slow times nor understaffed when busy.

Discrete Random Variable

A discrete random variable is like a list of specific outcomes that can be counted, such as the number of coins in your pocket or, as in our store's scenario, the number of customers arriving in a given time frame. Discrete stands in contrast to continuous, where outcomes can take on any value in a range.

In mathematics, discrete random variables are represented by whole numbers. This reflects real-world scenarios where you can't have 'half' a customer or 'three-quarter' of an email. The Poisson distribution used for the store example deals with discrete random variables because you can't have a fraction of a customer walk through the door. When working with any discrete random variable, equiprobability is not a necessity; that is, each outcome doesn't have to be equally likely. It's the totality of probabilities that must add up to one, reinforcing the certainty that one of the potential outcomes will occur. Understanding discrete random variables is crucial in fields like inventory management, quality control, and risk assessment, where the countable nature of occurrences is a central concern.