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 $E(X)$ or $\mu$. Represents the “average” (or expected) value.

Properties:

• If $c$ is constant, $E(c) = c$
• $E(X+c) = E(X) + c$
• $E(cX) = cE(X)$
• $E(g(x)) = g(E(x))$, where $g$ is a linear function

Median

Middle value dividing the distribution into 2 equal parts.

Variance

Denoted by $\text{Var}(X)$ or $\sigma^2$. Measures the spread or dispersion of the distribution around the mean.

Properties:

• $\text{Var}(c) = 0$
• $\text{Var}(X+k) = \text{Var}(X)$
• $\text{Var}(aX) = a^2\text{Var}(X)$
• $\text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X,Y)$
• If $X$ and $Y$ are independent, $\text{Cov}(X,Y) = 0$

Standard deviation

Square root of variance. Denoted by $\sigma$. $$

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 $\hat{F}$. Gives the proportion of observations less than or equal to $x$. Always non-decreasing.