The empirical distribution function, also known as the empirical cumulative distribution function, is a statistical tool used to estimate the underlying probability distribution of a given dataset. This function is particularly useful when working with large datasets where it may be difficult to calculate the exact probability distribution.

One example of the empirical distribution function is in the analysis of stock prices. Suppose we have a dataset of daily closing prices for a particular stock over the past year. To estimate the underlying probability distribution, we can construct the empirical distribution function by ordering the data from lowest to highest and then plotting the cumulative percentage of data points at or below each value. This plot can then be used to estimate the probability that the stock will close at or below a given value on a future day.

Another example is in the analysis of the distribution of exam scores for a particular class. Suppose we have a dataset of exam scores for all students in the class. To estimate the underlying probability distribution, we can construct the empirical distribution function by ordering the data from lowest to highest and then plotting the cumulative percentage of students who scored at or below each value. This plot can then be used to estimate the probability that a randomly selected student will score at or below a given value on a future exam.

In both of these examples, the empirical distribution function provides a useful way to estimate the underlying probability distribution without having to make assumptions about the specific form of the distribution. This can be particularly useful when working with complex or non-normal datasets. Additionally, the empirical distribution function can be easily updated as new data becomes available, allowing for more accurate estimates over time.