Distributions

In probability theory, a distribution refers to the way the values of a random variable are spread or allocated. It describes the likelihood or probability of different outcomes that the random variable can take. A distribution is defined by the probability that the random variable takes on each possible value (in the case of discrete variables) or falls within a certain range (in the case of continuous variables).

• Distributions
  • Random Variable and its Probability Distributions
  • Cumulative Distribution Function
  • Expectation of a Random Variable

Topics derived from distributions

✅ Probability Distributions Classification with Descriptions

These are distribution types in

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📊 Probability Distributions Table

Category	Distribution	Type	Description
1. Discrete	Bernoulli	Discrete	Models a single trial with two possible outcomes (success/failure).
	Binomial	Discrete	Models number of successes in n independent Bernoulli trials.
	Geometric	Discrete	Models trials until the first success.
	Negative Binomial	Discrete	Models trials until r successes occur.
	Poisson	Discrete	Models rare events in a fixed interval.
	Hypergeometric	Discrete	Models successes in dependent draws without replacement.
	Multinomial	Discrete	Generalizes Binomial to multiple outcomes.
	Zipf	Discrete	Models the frequency of elements ranked by popularity.
	Discrete Uniform	Discrete	All outcomes have equal probability.
	Categorical	Discrete	Models probabilities over multiple categories.
	Beta-Binomial	Discrete	Binomial distribution with Beta prior on the success probability.
	Zeta Distribution	Discrete	Models power-law behavior in discrete values.
2. Continuous	Normal (Gaussian)	Continuous	Bell-shaped curve, models natural phenomena.
	Exponential	Continuous	Models time until the next event occurs.
	Uniform	Continuous	All values within a range are equally likely.
	Gamma	Continuous	Models waiting time for multiple events.
	Beta	Continuous	Models probabilities of probabilities (fractions between 0 and 1).
	Log-Normal	Continuous	Models distribution of multiplicative processes.
	Chi-Square	Continuous	Models the sum of squared normal variables.
	Weibull	Continuous	Models time-to-failure and survival rates.
	Cauchy	Continuous	Heavy-tailed distribution.
	Laplace	Continuous	Symmetric with heavier tails than Normal.
	Rayleigh	Continuous	Models magnitude of a 2D vector with normally distributed components.
	Logistic	Continuous	Similar to Normal but with heavier tails.
	Pareto	Continuous	Power-law distribution, used in economics and social sciences.
	Gumbel	Continuous	Models the distribution of the maximum or minimum of samples.
	Frechet	Continuous	Models distribution of extreme events (heavy-tailed).
	Inverse Gamma	Continuous	Inverse of Gamma distribution, used in Bayesian inference.
	Multivariate Normal	Continuous	Generalizes Normal distribution to multiple variables.
	Multivariate T	Continuous	Generalization of T-distribution for multiple variables.
	Multivariate F	Continuous	Ratio of scaled multivariate chi-square distributions.
	Multivariate Beta	Continuous	Generalization of Beta distribution for multiple variables.
	Multivariate Gamma	Continuous	Generalization of Gamma distribution for multiple variables.
	Multivariate Chi-Square	Continuous	Sum of squared multivariate normal variables.
	Multivariate Inverse Gamma	Continuous	Inverse of Multivariate Gamma distribution.

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🔥 Key Differences

• Discrete vs. Continuous: Discrete distributions deal with countable outcomes, while continuous ones cover ranges of values.
• Multivariate vs. Univariate: Multivariate models multiple variables jointly, while univariate deals with a single variable.
• Heavy-Tailed Distributions: Cauchy, T-distribution, and Pareto have heavier tails than Normal, making them more prone to extreme values.

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References

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• date: 2025.03.17
• time: 12:20