Here below we write the density function (or the mass function) for some known random variables.
Normal distribution with mean \(\mu\) and standard deviation \(\sigma\)
\[f(x) = \frac 1 {\sigma \sqrt{2 \pi}} e^{- \frac{(x-\mu)^2}{2 \sigma^2}}\]
Uniform distribution on the real interval \([a,b]\)
\[f(x) = \left\{ \begin{array}{cc} \frac 1 {b-a} & if \, x \in [a,b] \\ 0 & otherwise \end{array} \right.\]
Exponential distribution of parameter \(\lambda\)
\[f(x) = \left\{ \begin{array}{cc} \lambda e^{-\lambda x} & x \geq 0 \\ 0 & x < 0 \end{array} \right.\]
Gamma distribution of parameter \(k\)
\[f(x) = \frac {x^{\alpha-1} \beta^\alpha e^{-\beta x }}{\Gamma(\alpha)}\]
Bernoulli distribution of parameter \(p\)
\[\mathbb{P}(X = 1) = 1 - \mathbb{P}(X = 0) = p\]
Binomial distribution of parameters \(n\) and \(p\)
\[\mathbb{P}(X = k) = \binom n k p^k(1-p)^{n-k}\]
Poisson distribution of parameter \(\lambda\)
\[\mathbb{P}(X = k) = \frac{\lambda^k}{k!} e^{-\lambda}\]