Introduction
It is widely known that pizza is one of the most highly demanded foods in State College.
In this case study, we analyze three different variables pertaining specifically to pizza sales. A probability mass function table was created using excel in order to illustrate the probability of a certain number of pizza sales occurring during a given hour. Since most pizza shops sell the highest volume of pizzas in the afternoon and into the evening, the probability measurements for each pizza shop were made between 2 P.M. and 8 P.M.
A number of observations can be made about the three data tables. The specific timeframe of pizzas sold between 2 P.M. and 8 P.M. was chosen because it is the busiest part of the day for most pizza shops in State College. Generally, 2 P.M. marks the start of any measurable pizza sales. Before that time, the amount of pizzas sold is scattered and difficult to measure. It is likely that the most amount of pizzas will be sold between 5 P.M. and 8 P.M. This timeframe is referred to in the industry as rush.
Description
In order to collect the data, a series of measurements of the amount of pizzas sold per hour over the course of one week were collected. A probability mass function table was then created using excel in order to illustrate the probability of a certain number of pizza sales for a given hour. In this probability mass function table, we let the variable X represent the number

of hours in a day, and the variable Y represent the number of pizza sales per hour. In this case, the value for hour 1 (X1) relates to 2P.M. on a given day, and X6 relates to 8P.M. The value of pizza 1 (Y1) relates to 1 pizza, and Y10 relates to 10 pizzas.
Along with the probability mass function relating to the probability of pizza sales per hour, the average price of a given pizza order for a given hour was also calculated. The purpose of this chart is to illustrate the true amount in sales of the specific pizza shop. Altogether, the probability of the amount of pizza sales per given hour, along with the average pizza cost per unit per hour, it is possible to determine the expected values of pizza sales per day during a sixhour shift.
Analysis
The data presented in this case study provides the probability of a certain number of pizzas being sold at a given time, and the average sale price per hour. Using the probability mass function data table, it was possible to solve for the expected values of X and Y, which would yield to the expected number of pizza sales per day. If this number is multiplied by the average sale price per day, it generates an estimate of the total amount in sales (in US dollars) during a six hour shift. With reference to the attached data sheet, since the expected value of X is 170.3, it can be expected that given the probabilities, approximately 170 pizzas will be sold during the six hour time period for any given day. Multiplying this number by the average sale price per pizza per shift, which was found to be 14.28, gives the expected amount of business revenue per six hour shift of $2,431.88. Based on experience through working at a pizza shop,

this number is an accurate number of business revenue between the hours of 2 P.M. and 8 P.M. on any given day.
In order to conduct a thorough investigation of the data, the variances, expected values, and correlations, and covariance’s between the PMF of the probability table were calculated using the formulas obtained throughout the lecture notes. This data is represented on the attached excel chart. The formulas used for data analysis are as follows:
Expected Values E[X] and E[Y] (This works for both X and Y):
Variance of X or Y (Formula is the same):
Covariance of X and Y:

Correlation of X and Y:
The following data table is a table representing the average sale prices of a pizza order per given hour:
As mentioned above, the average sale price per six hour shift multiplied by the amount of pizzas sold per day provides an estimate as to how much revenue is produced from pizza sales on a given day. As mentioned in the class lecture notes, the purpose of this analysis is to - probability of a number of events ("successes") occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event.
With this in mind, a formula representing the type of data collected in this assignment must be used in order to calculate more precise measurements of sale data. Given that the probability
(p) is known, the average or expected value of pizzas sold per day (E[X]) is known, and the average price of each pizza sold in a given hour is known, it is possible to use the Poisson

distribution to calculate the probability of a certain dollar amount of sales per day. The Poisson distribution formula is as follows:
Poisson Distribution PMF:
In this situation, if we let k = the amount of sales per day in dollars and λ = the probability of a number of pizza sales per given hour. Since there is already data pertaining to the average pizza sale price per hour, we could incorporate it into the formula in order to determine the probability of a given amount of sales per hour is. This formula could also be used to calculate the probability of a certain sales or revenue goal per six hour shift.
Conclusion
Determining the amount of sales revenue per day is an essential part of running a business efficiently. The data collected represents how the probability of a number of pizza sales per hour can be used to estimate the amount of sales revenue during six hours of operation. Although sales revenue is only part of the calculations required to analyze business efficiency, using the theories of probability distribution can yield to a better understanding of how much of a specific product to carry in store.