[DATA] A Random Process: Trains Your daily commute to work requires that you cross railroad tracks. At this particular railroad crossing the trains tend to be long and slow. So, getting stopped by a train will likely make you late for work. You start recording data to determine the likelihood of arriving at the tracks while a train is there. Open data set 5_1_37 at www.pearsonhighered.com/sullivanstats, which contains the day number and whether a train was present, or not, for 200 consecutive days in which you drove to work. The column “Train” shows a series of 0s and 1s. In that column, a 0 indicates there was no train present and a 1 indicates that a train was present. The column “Aggregate Train” represents the cumulative number of times a train was present. The column “Aggregate Proportion Train” represents the cumulative proportion of times a train was present.
f. What is the estimate of the probability of being stuck by a train during your commute?  

Verified step by step guidance

1. Identify the total number of days observed, which is given as 200 days in the data set.
2. Locate the column labeled 'Train' in the data set, where each entry is either 0 (no train) or 1 (train present).
3. Sum all the values in the 'Train' column to find the total number of days when a train was present during your commute.
4. Calculate the estimate of the probability of being stuck by a train by dividing the total number of days with a train present by the total number of days observed. Mathematically, this is expressed as: \[\text{Probability} = \frac{\text{Number of days with train present}}{\text{Total number of days}}\]
5. Interpret this ratio as the estimated probability that you will encounter a train during your commute on any given day.

Key Concepts

Probability Estimation

Probability estimation involves calculating the likelihood of an event occurring based on observed data. In this context, it is estimated by dividing the number of days a train was present by the total number of days observed. This relative frequency approach provides an empirical probability reflecting real-world outcomes.

Random Process

A random process is a sequence of random variables representing outcomes over time or space. Here, the presence or absence of a train each day is a random event, and the data collected over 200 days forms a random process that helps analyze patterns and probabilities of train arrivals.

Cumulative Proportion

Cumulative proportion is the running total of occurrences of an event divided by the total number of observations up to that point. It smooths out fluctuations in data and provides an ongoing estimate of probability, useful for understanding trends in the presence of trains over multiple days.