Discrete Random Variables¶

A random variable with a countable number of values

The number of variables could be either finite or infinite

Discrete Probability Distribution¶

Describes the probabilities of the occurrences of each possible outcome associated with a discrete random variable

Discrete uniform distributions: when all outcomes have the same possibility

Cumulative Probability Distributions for Discrete Random Variables¶

The probability that a discrete random variable is less than or equal to each of its possible values

Mean & Standard Deviation of A Discrete Random Variable¶

The weighted average of the possible values based on their probabilities, i.e. the expected value \(\mu_X\) or \(E(X)\)

\(\mu_X = \sum x_i \cdot P(x_i)\)

The variance of a discrete random variable is the expected value of the squared differences between the values and the mean

\(\sigma_X = \sqrt{\sum (x_i - \mu_X)^2 \cdot P(x_i)}\)

Linear Transformation of Random Variables¶

Definition¶

Every value of the variable is either multiplied by a constant or added to another constant or a combination of both

\(Y = a + bX\)

Linear transformation can be used to make the numbers more manageable

Affects on Mean & Standard Deviation¶

\(\mu_Y = a + b\mu_X\)

\(\sigma_Y = |b| \sigma_X\)

Affects on the Shape of Distribution¶

Addition: does not change
Multiplication: does not change if \(b > 0\), otherwise flipped horizontally

Linear Combinations of Random Variables¶

Multiples of given random variables are added together

\(Y = \sum\limits_{i=1}^n a_iX_i\)

Affects on Mean & Standard Deviation¶

\(\mu_Y = \sum\limits_{i=1}^n a_i\mu_i\)

\(\sigma_Y = \sqrt{\sum\limits_{i=1}^n a_i^2\sigma_i^2}\)