Distributions

A distribution describes how the values of a random variable are spread out. It provides a mathematical function that gives the probability of a random variable taking on each possible value.

Properties

Mean

Denoted by or . Represents the “average” (or expected) value.

Properties:

• If is constant,
• , where is a linear function

Variance

Denoted by or . Measures the spread or dispersion of the distribution around the mean.

Properties:

• If and are independent,

Covariance

Denoted by or . Measures the linear relationship between two random variables. Not standardized.

Properties:

Standard Deviation

Square root of the variance. Provides a measure of spread in the same units as the original random variable, making it often more interpretable than variance.

Distribution functions

Probability Mass Function

Used for discrete random variables. Gives the probability that a discrete random variable is exactly equal to some value .

For example, if represents the outcome of a fair die roll, then:

Probability Density Function

Used for continuous random variables. Doesn’t give the probability at an exact point (which is always 0 for continuous variables). Instead, gives the relative likelihood of the random variable taking on a value in a small interval around a point.

The probability of an event is the integral of the PDF over the region corresponding to the event:

For example, if follows a standard normal distribution, its PDF is

Cumulative Distribution Function

For both discrete and continuous random variables, the cumulative distribution function gives the probability that the random variable is less than or equal to . Always non-decreasing and has a range of .

For discrete random variables,

For continuous random variables,