probability density function

A continuous random variable is a function that assigns a real number to each element of a continuous sample space.

A continuous random variable \( X \) is said to follow a uniform distribution on the interval \(A = [a, b]\) if its probability density function is constant on that interval. Formally we have:

\[
U(x;a,b) =
\begin{cases}
\dfrac{1}{\\,b – a\\,} & x \in A \\\\[0.6em]
0 & x \notin A
\end{cases}
\]

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 gamma distribution is a continuous probability distribution defined on the positive half-line. Its probability density function is given by

\[
G(x;\alpha,\beta)=
\begin{cases}
\dfrac{1}{\beta^{\alpha}\\,\Gamma(\alpha)}\\, x^{\alpha – 1}\\, e^{-x/\beta} & x>0\\\\[10pt]
0 & \text{elsewhere}
\end{cases}
\]