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.

Parameters

Mean

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

Properties:

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

Median

Middle value dividing the distribution into 2 equal parts.

Variance

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

Properties:

• If and are independent,

Standard deviation

Square root of variance. Denoted by .

Skewness

Measure of the asymmetry about its mean. Positive skewness indicates a distribution with a longer right tail.

Kurtosis

Measure of the “tailedness” of the distribution. High kurtosis indicates a distribution with heavy tails or outliers.

Cumulative Distribution Function

Denoted by . Gives the proportion of observations less than or equal to . Always non-decreasing.