Variance of a Random Variable

The variance of a random variable is the variance of all the values that the random variable would assume in the long run. The variance of a random variable can be thought of this way: the random variable is made to assume values according to its probability distribution, all the values are recorded and their variance is computed. If this process is repeated indefinitely, the calculated variance of the values will approach some finite quantity, assuming that the variance of the random variable does exist (i.e., it does not diverge to infinity). This finite value is the variance of the random variable.

The variance of the random variable X is denoted by Var(X). For a discrete random variable, Var(X) is calculated as

Although this formula can be used to derive the variance of X, it is easier to use the following equation:

= E(x²) - 2E(X)E(X) + (E(X))²
= E(X²) - (E(X))²

The variance of the function g(X) of the random variable X is the variance of another random variable Y which assumes the values of g(X) according to the probability distribution of X. Denoted by Var[g(X)], it is calculated as

EX. The random variable X that assumes the value of a dice roll has the probability mass function:
p(x) = 1/6 for x Î {1, 2, 3, 4, 5, 6}.

=91/6 - 12.25
= 35/12

For any random variable X whose variance is Var(X), the variance of X + b, where b is a constant, is given by

Var(X + b) = E [(X + b) - E(X + b)]² = E[X + b - (E(X) + b)]²

= E [(X - E(X)]²
= Var(X)

i.e. the variance of a random variable does not change if a constant is added to all values of the random variable.

For any random variable X whose variance is Var(X), the variance of aX, where a is a constant, is given by

Var(aX) = E [aX - E(aX)]² = E [aX - aE(X)]²

= a²E[(X - E(X)]²
= a²Var(X)

The variance of a scalar function of a random variable is the product of the variance of the random variable and the square of the scalar.

For any two independent random variables X and Y, E(XY) = E(X) E(Y). Thus, the variance of two independent random variables is calculated as follows:

Var(X + Y) = E[(X + Y)²] - [E(X + Y)]²

=E(X² + 2XY + Y²) - [E(X) + E(Y)]²
=E(X²) + 2E(X)E(Y) + E(Y²) - [E(X)² + 2E(X)E(Y) + E(Y)²]
=[E(X²) - E(X)²] + [E(Y²) - E(Y)²]
= Var(X) + Var(Y)

Note that Var(-Y) = Var((-1)(Y)) = (-1)² Var(Y) = Var(Y). Therefore

Var(X - Y) = Var(X + (-Y)) = Var(X) + Var(-Y) = Var(X) + Var(Y)

The variance of the sum or difference of two independent random variables is the sum of the variances of the independent random variables. Similarly, the variance of the sum or difference of a set of independent random variables is simply the sum of the variances of the independent random variables in the set.