Definition:Bell-Shaped Curve

Definition

The following probability distributions exhibit what is known as a bell-shaped curve:

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $\Img X = \R$.

$X$ is said to have a Cauchy distribution if it has probability density function:

$\map {f_X} x = \dfrac 1 {\pi \lambda \paren {1 + \paren {\frac {x - \gamma} \lambda}^2} }$

for:

$\lambda \in \R_{>0}$

$\gamma \in \R$

This is written:

$X \sim \Cauchy \gamma \lambda$

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Then $X$ has a Gaussian distribution if and only if the probability density function of $X$ is:

$\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$

for $\mu \in \R, \sigma \in \R_{> 0}$.

This is written:

$X \sim \Gaussian \mu {\sigma^2}$