Pareto Distribution

Examples of Pareto Distribution in the following topics:

• Shapes of Sampling Distributions

  • The "shape of a distribution" refers to the shape of a probability distribution.
  • The shape of a distribution will fall somewhere in a continuum where a flat distribution might be considered central; and where types of departure from this include:
  • A normal distribution is usually regarded as having short tails, while a Pareto distribution has long tails.
  • Even in the relatively simple case of a mounded distribution, the distribution may be skewed to the left or skewed to the right (with symmetric corresponding to no skew).
  • Sample distributions, when the sampling statistic is the mean, are generally expected to display a normal distribution.

• Do It Yourself: Plotting Qualitative Frequency Distributions

  • Qualitative frequency distributions can be displayed in bar charts, Pareto charts, and pie charts.
  • Sometimes a relative frequency distribution is desired.
  • A special type of bar graph where the bars are drawn in decreasing order of relative frequency is called a Pareto chart .
  • This pie chart shows the frequency distribution of a bag of Skittles.
  • This graph shows the frequency distribution of a bag of Skittles.

• The Gauss Model

  • The normal (Gaussian) distribution is a commonly used distribution that can be used to display the data in many real life scenarios.
  • In probability theory, the normal (or Gaussian) distribution is a continuous probability distribution, defined by the formula:
  • A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.
  • If $\mu = 0$ and $\sigma = 1$, the distribution is called the standard normal distribution or the unit normal distribution, and a random variable with that distribution is a standard normal deviate.
  • Such variables may be better described by other distributions, such as the log-normal distribution or the Pareto distribution.

• Exploratory Data Analysis (EDA)

  • His reasoning was that the median and quartiles, being functions of the empirical distribution, are defined for all distributions, unlike the mean and standard deviation.
  • Moreover, the quartiles and median are more robust to skewed or heavy-tailed distributions than traditional summaries (the mean and standard deviation).

• t Distribution Table

  • T distribution table for 31-100, 150, 200, 300, 400, 500, and infinity.

• Student Learning Outcomes

• The Normal Distribution

  • Normal distributions are a family of distributions all having the same general shape.
  • The normal distribution is a continuous probability distribution, defined by the formula:
  • If $\mu = 0$ and $\sigma = 1$, the distribution is called the standard normal distribution or the unit normal distribution, and a random variable with that distribution is a standard normal deviate.
  • The simplest case of normal distribution, known as the Standard Normal Distribution, has expected value zero and variance one.
  • Many sampling distributions based on a large $N$ can be approximated by the normal distribution even though the population distribution itself is not normal.

• Chi Square Distribution

  • Define the Chi Square distribution in terms of squared normal deviates
  • The Chi Square distribution is the distribution of the sum of squared standard normal deviates.
  • The area of a Chi Square distribution below 4 is the same as the area of a standard normal distribution below 2, since 4 is 22.
  • As the degrees of freedom increases, the Chi Square distribution approaches a normal distribution.
  • The Chi Square distribution is very important because many test statistics are approximately distributed as Chi Square.

• The t-Distribution

  • Student's $t$-distribution (or simply the $t$-distribution) is a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.
  • The $t$-distribution with $n − 1$ degrees of freedom is the sampling distribution of the $t$-value when the samples consist of independent identically distributed observations from a normally distributed population.
  • The $t$-distribution was first derived as a posterior distribution in 1876 by Helmert and Lüroth.
  • Fisher, who called the distribution "Student's distribution" and referred to the value as $t$.
  • Note that the $t$-distribution becomes closer to the normal distribution as $\nu$ increases.

• Common Discrete Probability Distribution Functions

  • Your instructor will let you know if he or she wishes to cover these distributions.
  • A probability distribution function is a pattern.
  • These distributions are tools to make solving probability problems easier.
  • Each distribution has its own special characteristics.
  • Learning the characteristics enables you to distinguish among the different distributions.