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Probability Distribution - Continuous Functions／Models (Probability Density Functions)

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Continuous Probability Distributions
  • 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)

For all continuous variables, the probability mass function 𝑃𝑀𝐹(𝑥) is always equal to zero

𝑃𝑀𝐹(𝑥) = 𝐏(𝑋=𝑥) = 0 for all 𝑥

As a result, the 𝑃𝑀𝐹(𝑥) does not carry any information about a random variable 𝑋. Rather, we can use the cumulative distribution function 𝐶𝐷𝐹(𝑥)

  • 𝐶𝐷𝐹(𝑥) = 𝐏(𝑋≤𝑥)
  • 𝐶𝐷𝐹(𝑥) = 𝐏(𝑋<𝑥) + 𝐏(𝑋=𝑥)
  • 𝐶𝐷𝐹(𝑥) = 𝐏(𝑋<𝑥) + 0
  • 𝐶𝐷𝐹(𝑥) = 𝐏(𝑋<𝑥)

the derivative of a continuous 𝐶𝐷𝐹(𝑥) is a probability density function 𝑃𝐷𝐹(𝑥)

Continuous Probability Distributions - Calculating Statistics

see: Continuous Probability Distribution - Calculating Statistics

Continuous Probability Distributions - Types

Continuous Distributions

Description

Uniform Distribution
  • the probability is the same for every outcome in the sample space

Exponential Distribution
  • given the mean number of events per unit time (𝜆):
  • a Gamma variable = sum of 𝛼 independent Exponential variables

Gamma Distribution

Wishart Distribution

Normal Distribution
  • has a bell-shaped curve

Logistic Distribution
  • resembles the normal distribution in shape but has heavier tails

z-Distribution (Standard Normal Distribution)

t-Distribution

f-Distribution

Chi-Square Distribution
  • the sum of the squares of 𝑘independentstandard normal random variables

Chi Distribution

Maxwell-Boltzmann Distribution

Dirac Delta Distribution Function - Unit Impluse
  • is a probability distribution where all mass is around a single point

Beta Distribution
  • is a family of continuous probability distributions defined on the interval [0, 1] parametrized by two positive shape parameters, 𝛼 and 𝛽, that control the shape of the distribution

Multivariate Beta Distribution (MBD) - Dirichlet Distribution

Pareto Distribution (80-20 Rule)
  • is a skewed, heavy-tailed distribution that is sometimes used to model the distribution of incomes