Continuous Probability Distributions

used in scenarios where the set of possible outcomes is continuous (e.g. temperature on a given day)
ranges include:
- bounded intervals (a, b)
- unbounded intervals such as (a, +∞), (−∞, b), or (−∞, +∞)
- combinations of several such intervals
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): the number of events occurring within that unit of time has Poisson Distribution the time between events has an Exponential Distribution the total time of 𝛼 events has Gamma Distribution a Gamma variable = sum of 𝛼 independent Exponential variables
Gamma Distribution
Wishart Distribution	is a multivariate generalization of the gamma 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)	used as confidence multipliers
t-Distribution	used as confidence multipliers
f-Distribution	used as confidence multipliers
Chi-Square Distribution	the sum of the squares of 𝑘 independent standard normal random variables
Chi Distribution	the positive square root of the Chi-Square Distribution
Maxwell-Boltzmann Distribution	is a Chi Distribution with 3 degrees of freedom
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	is a generalization of Beta Distribution to multiple variables
Pareto Distribution (80-20 Rule)	is a skewed, heavy-tailed distribution that is sometimes used to model the distribution of incomes

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

Continuous Probability Distributions - Calculating Statistics

Continuous Probability Distributions - Types