Theorem 5.4.6. Alternate Formulas for Theoretical Measures.

  1. \(\displaystyle \sigma^2 = E[x^2] - \mu^2 = E[X(x-1)] + \mu - \mu^2\)
  2. \(\displaystyle \gamma_1 = \frac{1}{\sigma^3} \cdot \left [ E[X^3] - 3 \mu E[X^2] + 2\mu^3 \right ]\)
  3. \(\displaystyle \gamma_2 = \frac{1}{\sigma^4} \cdot \left [ E[X^4] - 4 \mu E[X^3] + 6\mu^2 E[X^2] - 3 \mu^4 \right ]\)

Proof.

in-context