Table of Contents

Enum DistributionType

Namespace
AiDotNet.Enums
Assembly
AiDotNet.dll

Represents different probability distributions used in statistical modeling and machine learning.

public enum DistributionType

Fields

Exponential = 4

A distribution that models the time between events in a process where events occur continuously and independently.

For Beginners: The Exponential distribution models the time between independent events that occur at a constant average rate.

Think of it as describing how long you wait for something to happen when the chance of it happening at any moment stays the same:

  • It's skewed to the right
  • Short waiting times are more likely than long ones
  • It has the "memoryless" property: the probability of waiting another hour doesn't depend on how long you've already waited

It's defined by one parameter:

  • ? (lambda): The rate parameter, which is the average number of events per unit time

Best used for:

  • Time between events in a Poisson process
  • Survival analysis
  • Reliability engineering
  • Waiting times

Examples:

  • Time between customer arrivals
  • Time until equipment failure
  • Length of phone calls
  • Time between radioactive decay events
  • Time until the next earthquake
Laplace = 1

A distribution with a sharper peak and heavier tails than the Normal distribution.

For Beginners: The Laplace distribution (also called the double exponential distribution) looks similar to the Normal distribution but has a sharper peak and heavier tails.

Think of it as a more "extreme" version of the Normal distribution:

  • It has a pointy center instead of a rounded one
  • It has fatter tails, meaning extreme values are more likely

It's defined by two parameters:

  • Location (µ): The center of the distribution
  • Scale (b): Controls how spread out the values are

Best used for:

  • Data with occasional large deviations
  • Error distributions in some financial models
  • Modeling sparse signals
  • Regularization in machine learning (L1 regularization)

Examples:

  • Financial returns with occasional large movements
  • Differences between paired observations
  • Error distributions in some regression problems
  • Modeling noise in signal processing
LogNormal = 3

A distribution whose logarithm follows a Normal distribution, resulting in a skewed shape.

For Beginners: The LogNormal distribution applies to data whose logarithm follows a Normal distribution.

Think of it as what happens when you take normally distributed data and exponentiate it (e^x):

  • It's always positive (never zero or negative)
  • It's skewed to the right (has a long tail on the right side)
  • Most values are clustered on the left side

It's defined by two parameters:

  • µ: The mean of the logarithm of the data
  • s: The standard deviation of the logarithm of the data

Best used for:

  • Quantities that are the product of many small independent factors
  • Data that can't be negative and is right-skewed
  • Growth processes

Examples:

  • Income distributions
  • House prices
  • Stock prices
  • Particle sizes
  • Length of time to complete a task
  • Many biological measurements
Normal = 0

The bell-shaped distribution that is symmetric around its mean (also known as Gaussian distribution).

The Normal distribution is the most commonly used probability distribution, characterized by its symmetric bell shape.

It's defined by two parameters:

  • Mean (µ): The center of the distribution
  • Standard deviation (s): How spread out the values are

Think of it as a distribution where:

  • Most values cluster around the mean
  • About 68% of values fall within 1 standard deviation of the mean
  • About 95% of values fall within 2 standard deviations
  • About 99.7% of values fall within 3 standard deviations

Best used for:

  • Natural phenomena that result from many small, independent factors
  • Measurement errors
  • Average values of large samples (due to Central Limit Theorem)
  • Many biological measurements (height, weight, etc.)

Examples:

  • Heights of people in a population
  • Measurement errors in scientific experiments
  • Test scores in large populations
  • Financial returns in efficient markets
Student = 2

A family of distributions that resemble the Normal distribution but have heavier tails (also known as t-distribution).

For Beginners: The Student's t-distribution (often just called t-distribution) looks similar to the Normal distribution but has heavier tails, meaning extreme values are more likely.

Think of it as a Normal distribution that's more forgiving of outliers:

  • It looks like a bell curve but with thicker tails
  • As the degrees of freedom increase, it gets closer to a Normal distribution

It's defined by one parameter:

  • Degrees of freedom (?): Controls the heaviness of the tails
    • Lower values = heavier tails
    • Higher values = more like a Normal distribution

Best used for:

  • Small sample statistics
  • Robust statistical methods
  • Data where outliers are expected
  • Confidence intervals when the population standard deviation is unknown

Examples:

  • Statistical tests with small samples
  • Financial returns modeling
  • Robust regression
  • Bayesian inference
Weibull = 5

Remarks

For Beginners: Probability distributions are mathematical functions that describe how likely different outcomes are.

Think of a probability distribution like a recipe for how values are spread out:

  • Some distributions create values clustered around a central point (like Normal)
  • Others spread values out differently (some have long tails, some are skewed)
  • Each has specific mathematical properties that make it useful for different situations

In machine learning, we use these distributions to:

  • Model uncertainty and randomness
  • Generate synthetic data
  • Make predictions with confidence intervals
  • Understand the underlying patterns in our data

Choosing the right distribution depends on what kind of data you're working with and what assumptions you can reasonably make about how that data is generated.

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