Probability Distribution

Distribution of a statistical data set (or a population) is a statistical function or listing that describes all possible values (or intervals) and their occurrence in a random experiment with it’s associated probability.

Probability Distribution Is a mathematical function that maps the all possible outcomes of an random experiment with it’s associated probability. It depends on the Random Variable X , whether it’s discrete or continues.

Types of Distributions:

• Continuous probability distributions: A continuous distribution describes the probabilities of the possible values of a continuous random variable. A continuous random variable is a random variable with a set of possible values (known as the range) that is infinite and uncountable.
  • Normal Distribution(Gaussian Distribution): It is a probability distribution with a bell-shaped curve where The peak always divides the distribution in half.
  • Exponential Distribution
  • Gamma Distribution
  • Chi-Square Distribution
  • Weibull Distribution
  • Laplace Distribution
  • Beta Distribution
  • T-Distributions
• Discrete probability distributions
  • Bernoulli Distribution: It describes a single experiment that has ONLY two outcomes.
  • Binomial Distribution: is a method of calculating probabilities for experiments having a fixed number of trials(successes in repeated Bernoulli experiments).
    • The binomial distribution is used to estimate the total number of successes from n trials when only two possible outcomes are there: success and failure.
    • The name Binomial suggests two mutually exclusive outcomes of trials.
  • Multinomial Distribution
  • Negative Binomial Distribution: The negative binomial distribution is similar to the Poisson distribution but with two parameters instead of one: r and p. In such a case, the Poisson distribution is the limiting case of the negative binomial distribution.
  • Geometric Distribution: It’s similar to the Binomial distribution, however, the experiment continues until the S successes are achieved
  • Poisson Distribution: Poisson distribution is used to model the number of occurrences of a certain event given a very large number of observations and the probability of the desired event to occur in each observation is significantly smaller.
  • Hypergeometric Distribution: The hypergeometric distribution is related to the number of successes in a sequence of N trials from a finite population without replacement.
  • Beta-binomial distribution
• Uniform Distribution: can be both Discrete and Continuous.
• Joint Probability Distribution
• Conditional Probability Distribution
• Data distribution types based on:
  • Number of peaks:
    • Unimodal distribution
    • Bimodal distribution
    • Multimodal distribution
  • Symmetry(Uniform): A symmetric distribution with little skewness which has two sides that are mirror images of each other and can have a peak(for normal distribution) or a bottom for U-shaped graphs.
  • Skewness: A measure of the deviation of a random variable’s given distribution(assymetry in the data or variable distribution) from the normal distribution.
    • Negative skew: Distribution Concentrated in the right, left tail is longer.
    • Positive skew: Distribution Concentrated in the left, right tail is longer.

Notes:

• Joint distributions: Getting a distribution over some combination of several random variables.
• Marginal distributions: If we have a joint distribution over some set of random variables, it is possible to obtain a distribution for a subset of them by “summing out” (or “integrating out” in the continuous case) the variables we don’t care about. I.e.

Reference:

• WikiPedia: List of probability distributions