Write a minimum of 250 words for each of the discussion questions below:

1. Select a discrete probability distribution, and provide a real-life example or application of that distribution. Explain how your example matches the conditions for the particular distribution that you have selected.

2. Select a continuous probability distribution, and provide a real-life example or application of that distribution. Explain how your example matches the conditions for the particular distribution that you have selected.

In your two replies to classmates, use reasoning to agree or to disagree with their classifications of the probability distributions that they have selected. Provide insights for ways to verify that their labeling of their selected probability distributions are appropriate.

Stu A

A probability distribution is a description of the different possible outcomes for a given random variable along with the likelihood of each of those outcomes (Evans, 2013). Probability distributions can either be discrete or continuous, so first I’ll focus on discrete. Discrete probability distributions are used to model a discrete random variable, which can be defined as a random variable whose outcomes are able to be counted (as oppose to *continuous *where the random variable spans across an interval of measurement, as the individual increments in between are too numerous to count). A Poisson distribution is a type of discrete probability distribution which models the number of occurrences of a specific outcome over a given interval of time or some other unit of measure (Evans, 2013). An example that comes to mind from real-life is a model of the number of traffic violations from vehicles passing through a red light at a certain intersection over a week’s time. This distribution is discrete as there can only be complete violations counted , so there are no decimals included and the number is finite. This example satisfies all three criteria of a Poisson distribution. First, there is no assumption as to any limit on the number of violations. Secondly, each violation occurring is independent from another, since one car passing through a red doesn’t affect any other cars (generally speaking). And lastly, the average number of violations, or the *rate,* is a constant as is generally the case according to police records.

A continuous probability distribution is used to model a continuous random variable which is a random variable with continuous outcomes over any interval of real numbers, such as units of measurement, like time, weight, temperature, height, etc. The exponential distribution is an example of a continuous probability distribution and is used to model the amount of time that elapses between randomly occurring events (Evans, 2013). The Poisson distribution is very much associated to the exponential distribution, since they both only have one parameter (the average rate or *lambda*) and that the exponential distribution can always be used to estimate the time lapses between occurrences that are estimated by the Poisson distribution. It therefore follows, that the amount of time that passes between red-light traffic violations can be predicted using an exponential distribution. One criteria for the exponential distribution is that it is *memoryless, *which means that the current time lapse since the last outcome does not affect the amount of time that elapses until the next outcome. This property is again satisfied in the instance above with the red-light traffic violations, as the time until the next violation will occur is not affected by how much time already elapsed from the most recent violation.

Reference

Evans, J. R. (2013). *Statistics, data analysis, and decision modeling *(5th ed.). Upper Saddle River, NJ. Pearson Education.

STU B

The discrete probability distribution that is used in real life is to analyze the number of car accidents that had happened per month in Dhaka (Banik, 2009). For that part, the accidents are calculated in a Poission distribution. In that scenario, there are two directions that are calculated where one is in the negative and one is in the positive direction to determine which month will have the most accidents in a specific location in the city. For that method, it is studied in a select amount of years that will be used to study, say a five to six year sequence to identify if there are patterns in a certain month given on a certain year. The study will determine if there are any observations on specific conditions such as the driver’s actions, particularly in the skill level and awareness. However, there are more than just one condition that is studied with the Poisson such as using the weather or road conditions to look at which months will cause more accidents on a certain year. The model uses zero-inflated values in order to identify if there are any variances in the model. The study identifies the different trials to determine the outcome whether it is random or not. The number of n can be large as there can be a large amounts of traffic traveling during the day throughout the selected amount of time that is identified.

Continuous probability distribution is used in a real life example is to observe the prices of the stock changes overtime (Pareek, 2009). The idea with the stock prices always have various changes throughout the day from opening to close. The idea with the stock prices indicate that the prices is not always exact to what it is displayed. For that scenario, the cumulative density function plot is created (CDF) where it can be modeled to identify where it will show the continuous price of a certain stock. The binomial distribution is another method that can be used in order to determine if the stock is either a profit or a loss for its distribution. There is a prediction on whether or not there will be either a success or fail by also using the probability density function in order to determine if the stock will perform better than the competitor. Predicting stocks does show the similarity on flipping the coin to predict the outcome whether it will be heads or tails after a set amount of attempts. Just as with the stock prices, there are a number of attempts on whether or not the stock will make a profit based on how long have the stockholder have bought it and when is the right time to sell it in order to generate the profit. In addition, the continuous probability can be unpredictable when it comes to stock investments.

Reference:

Banik, S. & Kibria, B.M. (2009, Nov. 1). “On Some Discrete Distributions and their Applications with Real-Life Data.” Wayne State University. Retrieved from https://digitalcommons.wayne.edu/cgi/viewcontent.cgi?article=1518&context=jmasm

Pareek, M. (2009, Oct. 21). “Distributions in Finance.” Risk Prep. Retrieved from https://www.riskprep.com/all-tutorials/36-exam-22/63-distributions-in-finance