Discrete random variance

The variance is, in short, a measure of how much the values of a random variable differ from the mean. It’s the square of the standard deviation.

Notation

We write the variance of a list of values X as Var(X).

Suppose we have a list of possible discrete values and their probabilities.
The value at position n is x_n and the associated probability is p_n.

The variance of the list is calculated by:

Var(X) = E(X^2) - (E(X))^2

We can substitute the formula for expectation into this - E(X) = \sum x_n p_n - to get:

Var(X) = \sum x_n^2 p_n - \left( \sum x_n p_n \right)^2

Question	Answer
Var(X) = E(X²) - (E(X))²	The formula for variance of a discrete random variable is Var(X) = E(X^2) - (E(X))^2.
What is the expanded formula for Var(X) using summation?	Var(X) = \sum x_n^2 p_n - \left( \sum x_n p_n \right)^2, where x_n is a value and p_n is its probability.
What does the variance measure?	The variance measures how much the values of a random variable differ from the mean.
How is variance related to standard deviation?	Variance is the square of the standard deviation.
What is the notation for the variance of a list of values X?	Var(X)