# Discrete Random Variables: Probability distributions

## Discrete Random Variables

• A discrete random variable is a variable that can take on a finite or countable number of values.
• Each possible value of the discrete random variable can be associated with a probability.
• The outcomes of a discrete random variable are obtained from a random process or experiment.

## Probability Distribution of Discrete Random Variables

• The probability distribution of a discrete random variable is a list of probabilities associated with each of its possible values.
• It usually takes the form of a probability histogram.
• Probability distributions are typically denoted by p(x), where x is the possible outcome.
• For discrete random variables, the cumulative distribution function can also be defined.

## Characteristics of Probability Distribution

• For all outcomes x, 0 ≤ p(x) ≤ 1. Probabilities are always between 0 and 1, including 0 or 1.
• The sum of all probabilities for all possible values of x equals 1. This is known as the normalizing condition and can be expressed as Σp(x) = 1.

## Expectation and Variance

• The expected value (mean) of a discrete random variable X is a weighted average of all possible values of X, with weights given by their respective probabilities, denoted as E(X).
• The variance of a discrete random variable X is the average of the squared differences from the mean, given by the formula Var(X) = E[(X-E(X))^2].
• The standard deviation is the square root of the variance, given by SD(X) = √Var(X).

## Applications

• Discrete random variables and their distributions are valuable tools in predictive modelling, risk assessment, and decision-making processes in statistics.
• Some common discrete probability distributions are the binomial distribution, Poisson distribution, and geometric distribution.