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¶

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

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

Independence's Effect on Combination¶

Mean of Difference¶

The mean of the difference between two variables is always the difference of their means, regardless of whether they are independent or dependent. This is a property of the expected value:

$\mu_{X-Y} = \mu_X - \mu_Y$

Standard Deviation of Difference¶

The formula for the variance of the difference between two dependent variables is:

$\sigma_{X-Y}^2 = \sigma_X^2 + \sigma_Y^2 - 2 \cdot r \cdot \sigma_X \sigma_Y$

Where $r$ is the correlation coefficient between X and Y.

If the variables were independent, $r$ would be 0, and we could calculate the standard deviation.
Because they are not independent, we need to know the correlation $r$ to calculate the exact standard deviation.

Discrete Random Variables¶