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

Written by

Sahithyan K.

September 29, 2026 1 min read

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