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Univariate Probability Distribution

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univariate probability distribution (sometimes just called probability distribution)

  • is a model that describes the probability of a random variable
  • is essentially a “list” of all possible outcomes (of the random variable) along with their corresponding probability value
  • a variety of phenomena can be described by relatively few families of probability distribution models

UPD - Various Function Forms

  • probability distribution function (PDF) - is a function used to describe the probability distribution of a random variable
    • probability mass function (PMF) - is a probability distribution function that describes a DISCRETE random variable
    • probability density function (PDF) - is a probability distribution function that describes a CONTINUOUS random variable
  • cumulative distribution function (CDF) - is the integral of the probability distribution function
  • reverse cumulative distribution function (RCDF) or survivor distribution function (SDF) -
  • hazard distribution function (HDF) -
  • cumulative hazard distribution function (CHDF) -
  • quantile function or inverse cumulative distribution function (ICDF) -
  • moment generating function (MGF) of 𝑋 -
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UPD - Modeling a Random Variable (Discrete or Continuous)

The univariate probability distribution of a random variable is either discrete or continuous:

Random Variable Type

Description

discrete probability distribution

aka

probability mass function

  • used in scenarios where the set of possible outcomes are discrete, either:
    • a finite number of values (e.g. coin toss or dice roll)
    • an infinite sequence of values (e.g. counting numbers)
  • encoding a discrete list of the probabilities of the outcomes
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continuous probability distribution

aka

probability density function

  • used in scenarios where the set of possible outcomes is continuous (e.g. temperature on a given day)
  • ranges include:
  • the probability of any individual outcome equals zero (it’s possible, it’s just probability zero)
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why Probability Distribution - Density vs Mass?

Distribution Comparisons

see: Probability Distribution - UPD Comparisons

Distribution Properties/Theorems/ Computing Variate Analysis Characteristics

Combining Multiple UPDs

see: Multivariate Probability Distribution