Distribution measures for discrete random variables

Variance

The variance of a metric discrete random variable X with support \(T = {x_1, x_2, \dots, x_k, \dots}\), probability distribution \(p_1, p_2, \dots, p_k, \dots\) and expected value is given by

\[o^2 = Var(X) = \sum_{i \ge 1} (x_i - \mu)^2p_i\]

The variance corresponds to the expected squared deviation of the random variable X from its expected value
Alternative Formula:

\[o^2 = Var(X) = E(X-\mu)^2\]

The standard deviation is the square root of the variance: \[o = +\sqrt{Var(X)}\]

Estimators: \(\tilde{S^2} = g(X_1, \dots, X_n) = \frac{1}{n}\sum_{i = 1}^{n}(X_i - \overline{X})^2\) and \(S^2 = g(X_1, \dots, X_n) = \frac{1}{n - 1}\sum_{i = 1}^{n}(X_i - \overline{X})^2\)

Confidence interval: \([\overline{X}-t_{1-\alpha/2}(n-1) * \frac{S}{\sqrt{n}}, \overline{X}+t_{1-\alpha/2}(n-1) * \frac{S}{\sqrt{n}}]\)

Frequency interpretation of the variance

• Var(X) can be interpreted as the average squared deviation from the arithmetic mean that results when we repeat the random process often enough

Properties of the variance

• Displacement rule: \(Var(X) = E(X^2) - (E(X))^2 = E(X^2) - \mu^2\)
• For independent metric random variables X and Y: \(Var(X+Y) = Var(X) + Var(Y)\)
• For independent metric random variables X and Y and arbitrary real numbers \(a_1, \dots, a_n\): \(Var(a_1X_1 + \dots + a_nX_n) = a_1^2Var(X_1) + \dots + a_n^2Var(X_n)\) Back to Table of Contents