Common Probability Distributions

A Probability distribution list the probabilities of all possible outcomes of a random variable. It quantifies the likelihood of each possible values.

Types of Probability Distributions

  1. Discrete Probability Distribution: Applies to discrete random variables (those with finite outcomes). It provides the probability for each possible value of the random variable. e.g., throw a dice or pick a card.

  2. Continuous Probability Distribution: Applies to continuous random variables (those with an infinite number of outcomes within a range). It describes the probability of the random variable falling within a particular interval. e.g., measurement of human height.

Notation

  1. Random Variable (X, Y, Z, ...): A variable representing the outcomes of a random phenomenon. It's usually denoted by uppercase letters like X, Y, Z.
  2. Probability Mass Function (PMF) - P(X=x): For a discrete random variable X, the PMF gives the probability that X is exactly equal to some value x. It's denoted as P(X=x) or p(x).
  3. Probability Density Function (PDF) - f(x): For a continuous random variable X, the PDF describes the probability of X falling within a differential interval around x. The PDF is denoted as f(x) and is such that the integral of f(x) over the entire space is 1.
  4. Cumulative Distribution Function (CDF) - F(x): The CDF gives the probability that a random variable X is less than or equal to a certain value x. It's denoted as F(x) and defined for both discrete and continuous random variables.
  5. Expected Value (Mean) - E(X): The mean or expected value of a random variable X is the long-run average value of repetitions of the experiment it represents. It's denoted as E(X) or μ.
  6. Variance - Var(X): The variance of a random variable X is a measure of how much values in the distribution vary on average from the mean. It's denoted as Var(X) or σ².
  7. Standard Deviation - σ(X): The standard deviation is the square root of the variance. It provides a measure of the spread of the distribution around the mean.