Discrete Distribution

For all the definition below:

• $N$ is the population size.
• $n$ is the sample size.

Parameters

Mean

Average of the all values in the entire population. Usually unknown. Denoted by $\mu$. $$

$$\mu = \frac{1}{N} \sum_{i=1}^N x_i$$

Variance

Measure of the spread of the observed values. Denoted by $\sigma^2$. $$

$$\sigma^2 = \frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2$$

Standard Deviation

Square root of the variance. Denoted by $\sigma$. $$

Skewness

$$\gamma_1 = \frac{\mathbb{E}[(X - \mu)^3]}{\sigma^3}$$

Kurtosis

$$K=\frac{\mathbb{E}\left[ (X-\mu)^4 \right]}{\sigma^4}$$

Sample Parameters

Sample mean

Average of the observed values. Denoted by $\bar{x}$. $$

$$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$$

Sample variance

Denoted by $s^2$. $$

$$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$$

Here sample mean $\bar{x}$ is used instead of $\mu$, which underestimates the variance. $n-1$ is used instead of $n$ to correct this bias, which is called Bessel’s correction. This correction makes it an unbiased estimator.

Sample covariance

$$\mathrm{cov}(X, Y) = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})$$

Sample skewness

$$g_1 = \frac{n}{(n-1)(n-2)} \sum_{i=1}^n \left(\frac{x_i - \bar{x}}{s}\right)^3$$