Probability Distributions

A probability distribution is a function that describes the likelihood of obtaining the possible values that a random variable can assume. In other words, the values of the variable correspond to outcomes of a random phenomenon.

Discrete Probability Distributions

Discrete probability distributions apply to the scenarios where the set of possible outcomes is discrete. Common discrete probability distributions include:

  • Bernoulli Distribution: It has only two possible outcomes, success (1) and failure (0).

  • Binomial Distribution: It models the number of successes in a fixed number of Bernoulli trials with the same probability of success.

  • Poisson Distribution: It models the number of events happening in a fixed interval of time or space.

Continuous Probability Distributions

Continuous probability distributions apply to scenarios where the set of possible outcomes can take on values in a continuous range. Common continuous probability distributions include:

  • Uniform Distribution: All outcomes are equally likely within a certain range.

  • Normal Distribution: Also known as the Gaussian distribution, it's a bell-shaped curve where the mean, median, and mode are all the same.

  • Exponential Distribution: It models the time between events in a Poisson process.

Probability Density Function and Cumulative Distribution Function

For a continuous probability distribution, the probability density function (pdf) defines the probability that the random variable takes on a specific value. Since it's continuous, the probability at a specific point is technically zero; we instead look at the probability within a certain range.

The cumulative distribution function (cdf) of a random variable is defined as the probability that the variable takes a value less than or equal to a certain value.

Expectation and Variance

The expected value (or mean) of a random variable is the long-run average value of repetitions of the experiment it represents.

The variance of a random variable is a measure of how much the values of the random variable vary around the expected value. The standard deviation is the square root of the variance.