In an essay of no less than three pages, contrast the major differences between the normal distribution and the exponential and Poisson distributions. Include an example situation where each one is best suited to be matched to answer a question. Be sure to provide research to support your ideas. Use APA style, and cite,in-text cite and reference your sources to avoid plagiarism.

The Exponential Distribution The natural logarithm e is actually about = 2.71828. It is a relationship a lot like π for circles and spheres in that e reflects how, when matched to a variable such as x, a natural logarithmic function goes to infinity as x increases, and to negative infinity as x goes to 0. For this to be true, e e = e 0 = 1. You can be as glad that someone else figured this out as you are that others figured through mathematics where e goes in probability distributions to help calculate things needed to find in two situations: where Exponential Distributions and Poisson Distributions can help us (Wolfram MathWorld, 2016). The Exponential Distribution was developed from e to solve queuing questions, and as you see in the textbook, most often how long something or someone can receive a certain service if the business has a certain capacity. The Exponential Distribution has this formula: F(X) = μe -μx Where X = random variable, e.g. service times μ = average number of units (e.g., services the facility can do in a given time) e = natural logarithm constant = 2.718

You may recognize the probability, which once again will be the area under the curve from one limit of X to the other, has a logarithmic range and all the probability/area under that logarithmic curve = 1. Mathematical truths mean these equations are also true, and you can use them: Expected value = 1/μ = e.g., average service time Variance = 1 / μ2 This also means the probability that X is either less than or equal to a time t is: P(X ≤ t) = 1 – e -μt The Arnold’s Muffler example (page 49 of the textbook) was a good one of the Exponential Distribution, as it featured service times, and the question of what the probability would be of the installation of a new muffler being 0.5 hours or less was the area under the above curve from X (or t) = 0 to X (or t) = 0.5.

The Poisson Distribution The Poisson Distribution is also good for determining services in given times, and just for discrete probability distribution, which is why the probability of customer arrivals at a bank was a good example in the textbook (page 50). The probability of exactly X arrivals or occurrences is P(X), and P(X) = λ xe -λ X! Where λ = the mean arrival rate (e.g., the average number of arrivals in a given unit of time) e = natural logarithm constant = 2.718 X = number of occurrences For discrete Poisson Distributions, Expected value = λ and Variance = λ.

Notice that when the Expected Value = mean = λ = 2, as in two arrivals to a bank per hour, the distribution peaks at one to two arrivals. On the graph, you can see that at hardly any time do seven or more people arrive in an hour. This number is of low occurrence and probability even when λ = 4, and the curve as represented by discrete bars (no “half-people”) shifts to the right to higher numbers of X. Subjective probability determination (e.g., “I don’t think that’s going to happen”) may seem hasty and shallow in research, but the speaker may have years of experience in the topic at hand. From these four units, you know objective probability assessments. Thanks to mathematics, an easy and discrete approach is almost reflexively resorted to when one is asked about a coin toss with two outcomes possible. Or you may see that the probability distribution (the range of probabilities) are in fact continuous (e.g., the probability that a contractor will perform under a given amount of days), and the probability is the area under the continuous probability curve between the limits, or one limit and 0 as calculated by the formulas. One can, and has, made societies better by learning mathematical truths and leveraging their realities with probability approaches. Statisticians support businesses and governments by choosing and defending the choice of a probability distribution approach, performing the research for variable amounts and calculations, and presenting the solution for decision. In the second half of this course, you will focus more on the leadership aspect of quantitative analysis: the decision-making process and challenges posed by probabilities, and how to enhance leadership decision-making based on business analysis.

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Render, B., Stair, R. M., Jr., Hanna, M. E., & Hale, T. S. (2015). Quantitative analysis for management (12th ed.). Upper Saddle River, NJ: Pearson.