Week 6 - Continuous Probability Distributions

Week 6 - Continuous Probability Distributions

Objectives

  • Understand the properties of continuous probability distributions.
  • Understand the uniform distribution.
  • Understand the exponential distribution.
  • Understand the normal distribution.

Continuous Probability Distributions

  • Review of the difference between discrete and continuous data.
  • Focus on the distributions of continuous data.

Examples of Continuous Data

  • Amount of rainfall in inches in a year for a city.
  • Weight of a newborn baby.
  • Height of a randomly selected student.

Properties of Continuous Probability

  • A random variable X is continuous if possible values comprise either a single interval on the number line or a union of disjoint intervals.
  • Example: In the study of the ecology of a lake, X (the random variable) may be depth measurements at randomly chosen locations. X is a continuous random variable. The range for X is the minimum depth possible to the maximum depth possible.

Properties of Continuous Probability (Continued)

  • In principle, variables such as height, weight, and temperature are continuous, but measurement limitations restrict us to a discrete (though finely subdivided) world.
  • Continuous models often approximate real-world situations very well.
  • Continuous mathematics (calculus) is frequently easier to work with than mathematics of discrete variables and distributions.

Properties of Continuous Probability (Graphs)

  • The graph of a continuous probability distribution is a curve.
  • Probability is represented by the area under the curve.
  • Area under the curve is given by the cumulative distribution function (cdf).
  • The cumulative distribution function is used to evaluate probability as area.

Properties of Continuous Probability (Cumulative Distribution Function)

  1. The outcomes are measured, not counted.
  2. The entire area under the curve and above the x-axis is equal to one.
  3. Probability is found for intervals of x values rather than for individual x values.

Uniform Distribution

  • The uniform distribution is a continuous probability distribution concerned with events that are equally likely to occur.
  • Note if the data is inclusive or exclusive of endpoints.

Uniform Distribution (Detailed)

  • The uniform distribution is a continuous random variable with equally likely outcomes over the domain a < x < b.
  • It is often referred to as the rectangular distribution because the graph has the form of a rectangle.

Exponential Distribution

  • Exponential distributions provide probability models widely used in engineering and science disciplines.
  • One important application is to model the distribution of lifetimes or times to an event.

Exponential Distribution (Memoryless Property)

  • A partial reason for the popularity is the "memoryless" property of the Exponential distribution.
  • Future probabilities do not depend on any past information.

Normal Distribution

  • The Normal Distribution is a family of continuous distributions that can model many histograms of real-life data which are mound-shaped (bell-shaped) and symmetric (for example, height, weight, etc.).
  • It is probably the most important distribution in all of probability and statistics.
  • Many populations have distributions that can be fit very closely by an appropriate normal (or Gaussian, bell) curve.

Normal Distribution (Parameters)

  • The mean can be any real number, and the standard deviation is greater than zero.
  • The normal curve ranges from negative infinity to infinity.