Basic Concepts

  • Probability is a measure of the likelihood of an event occurring.
  • It is quantitatively represented as a number between 0 and 1, inclusive.
  • The probability of an impossible event is 0, while the probability of a certain event is 1.
  • An experiment is a process that leads to one of several possible outcomes. The set of all possible outcomes is the sample space.
  • An event is a subset of the sample space.

Probability Axioms

  • Nonnegativity: The probability of an event is always a nonnegative real number.
  • Normalization: The probability of the sample space is 1.
  • Additivity: The probability of the union of two mutually exclusive events is the sum of their probabilities.

Probability Laws

  • Complementary rule: The probability of the complement of an event is 1 minus the probability of the event.
  • Multiplication rule: The probability of the intersection of two events is the product of the probabilities of the events, provided the events are independent.
  • Addition rule: The probability of the union of two events is the sum of their probabilities minus the probability of their intersection.

Conditional Probability

  • Conditional probability is the probability of an event given that another event has occurred.
  • The formula for conditional probability is **P(A B) = P(A ∩ B) / P(B)**, where **P(A B)** denotes the probability of event A given that event B has occurred.

Bayes’ Theorem

  • Bayes’ theorem provides a way to revise existing predictions or theories given new or additional evidence.
  • The theorem is expressed as **P(A B) = [P(B A) * P(A)] / P(B)**.

Random Variables

  • Random variables are numerical outcomes of a random phenomenon.
  • A discrete random variable can take on a countable number of values, while a continuous random variable can take on an infinite number of values within an interval.

Probability Distributions

  • A probability distribution describes how probabilities are distributed over the values of a random variable.
  • For discrete random variables, this distribution can be described by a probability mass function. For continuous random variables, it can be described by a probability density function.

Expected Value and Variance

  • The expected value (mean) of a random variable is essentially an average of the possible outcomes, with each outcome weighted by its probability.
  • The variance measures how far each number in the set is from the mean and thus from every other number in the set.
  • The standard deviation is the square root of the variance and provides a measure of the average distance from the mean.

Important Distributions

  • Some important distributions for continuous random variables are the uniform distribution, the normal distribution, and the exponential distribution.
  • Important distributions for discrete random variables include the binomial distribution, the Poisson distribution, and the geometric distribution.