Contents

Probability Distributions

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Some Common Distributions

Normal (and Log-Normal) Distribution

The normal distributions are used to explain central tendencies

  • Defined using two parameters

    • Mean (\(\mu\)) and Standard Deviation (\(\sigma\))

    • Symmetric distribution

    • Represents additive processes

  • The lognormal distribution stipulates that the log of random variable x (i.e., log(x) is normally distributed

    • Does not work for negative data (in orginal space)

    • Represents multiplicative process

Lognormal distribution is often appropriate when the data are skewed. (See the Gamma distribution later on) Normal and Lognormal Distributions are best suited to represent central tendencies

Python Examples

Gamma Distribution

Python Examples

Poisson Distribution

  • A discrete distribution used to represent the number of independent events within a fixed time

    • Number of independent rainfall events within a year

  • Related to Exponential Distribution

  • Continuous distribution for inter-arrival times

  • Poisson distribution assumes stationarity

  • Rate at which events occur is constant

An example of application is Synthetic Rainfall Generation Model for Evaluating Potential Erosion at Highway Construction Sites

Python Examples

Exponential Distribution

  • Often used to model the time between two independent events

    • Time between two rainstorms

  • Exponential distribution is represented by the parameter (\(\lambda\))

    • Reciprocal of Average inter-event time

  • Related to Poisson Distribution

  • A discrete distribution for number of events in a fixed time

Python Examples

Binomial Distribution

Used when there are two outcomes

  • Success and Failure

  • The probability of success for each event is denoted by “p”

    • Often assumed stationary

  • The PMF calculates the probability of success of x out of N total events

The binomial distribution is often used for risk and reliability calculations

Python Examples

Logistic Distribution

Python Examples

Multinomial Distribution(s)

Python Examples

References

  1. Chan, Jamie. Machine Learning With Python For Beginners: A Step-By-Step Guide with Hands-On Projects (Learn Coding Fast with Hands-On Project Book 7) (p. 2). Kindle Edition.

  2. Machine Learning for CE Probability Distributions (Fall 2020 Lesson)

  3. Synthetic Rainfall Generation Model for Evaluating Potential Erosion at Highway Construction Sites