Discrete Random Variables¶

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¶

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

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