normal distribution

The Gaussian function is a bell-shaped function defined by an exponential quadratic expression, widely used to represent the normal distribution in probability and statistics.

The Beta distribution is a continuous probability distribution defined over the open interval \( (0, 1) \). In formal terms, the beta distribution is defined by the following probability density function:

\[
B(x; \alpha, \beta) =
\frac{x^{\alpha – 1}(1 – x)^{\beta – 1}}{B(\alpha, \beta)}
\quad 0 < x < 1 \]

The normal distribution is a continuous probability distribution defined by its mean \(\mu\) and standard deviation \(\sigma\), forming a symmetric bell-shaped curve. the normal probability density function is:
\[
\mathcal{N}(x; \mu, \sigma) = \frac{1}{\sqrt{2\pi} \\,\sigma} \\, e^{-\frac{1}{2\sigma^2}(x – \mu)^2}
\]

The Standard Normal Z Table shows the cumulative probability for each z value, helping to find how likely a standard normal variable is less than or equal to a given z-score.

The Student’s t distribution models how a sample mean behaves when the population variance is unknown and estimated from data. It is defined as:

is defined as:

\[
T = \frac{Z}{\sqrt{V/k}}
\]

A sampling distribution represents the distribution of a statistic obtained from all possible samples of a given size drawn from a population.