# Probability distributions for general discrete random variables

## Probability distributions for general discrete random variables

## Understanding Discrete Random Variables

**Discrete Random Variables**are those which can take only specific discrete values. For example, the number of heads obtained when flipping a coin multiple times.- In contrast,
**Continuous Random Variables**can take all values in a continuous range. - Each specific outcome of a discrete random variable is assigned a
**probability**, which must sum to 1 across all possible outcomes.

## Probability Mass Function (PMF)

- The
**Probability Mass Function**(PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value. - It must satisfy two conditions:
- The PMF is always non-negative i.e. P(X=x) >= 0
- The sum of all probabilities for all possible outcomes is equal to 1 i.e. ∑ P(X=x) = 1

## Cumulative Distribution Function (CDF)

- 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. - The CDF is a non-decreasing function, i.e., if x1 <= x2 then F(x1) <= F(x2)
- For discrete random variables, CDF is a step function.

## Expectation and Variance

- The
**Expectation**or the expected value of a discrete random variable gives the long-run average value of the variable. It is calculated by summing the products of each possible value the random variable can take and its corresponding probability. - The
**Variance**of a discrete random variable measures the spread of the probability distribution. It is calculated by taking the average of the squared deviations from the expected value. - The
**Standard Deviation**is the square root of the variance, providing a measure of spread that is in the same units as the random variable itself.

## Working with multiple discrete random variables

- When considering two discrete random variables together, one must consider both the
**Joint Probability Mass Function**(joint PMF) and the**Marginal Probability Mass Function**(marginal PMF). - The joint PMF gives the probability of each possible pair of outcomes, while the marginal PMF gives the probabilities for each variable considered separately.
- If the outcome of one variable does not affect the outcome of the other, the variables are said to be
**Independent**. Here, the joint PMF is the product of the marginal PMFs.

Remember to practice plenty of past papers and mark them thoroughly for a deeper understanding. Working through examples and problems is key to understanding and mastering these concepts.