.. _discrete-randint:

Discrete Uniform (randint) Distribution

The discrete uniform distribution with parameters :math:\left(a,b\right) constructs a random variable that has an equal probability of being any one of the integers in the half-open range :math:[a,b). If :math:a is not given it is assumed to be zero and the only parameter is :math:b. Therefore,

.. math:: :nowrap:

\begin{eqnarray*}
    p\left(k,a,b\right) & = & \frac{1}{b-a} \quad a \leq k < b \\
    F\left(x;a,b\right) & = & \frac{\left\lfloor x\right\rfloor -a}{b-a} \quad a \leq x \leq b \\
    G\left(q;a,b\right) & = & \left\lceil q\left(b-a\right)+a\right\rceil \\
    \mu & = & \frac{b+a-1}{2}\\
    \mu_{2} & = & \frac{\left(b-a-1\right)\left(b-a+1\right)}{12}\\
    \gamma_{1} & = & 0 \\
    \gamma_{2} & = & -\frac{6}{5}\frac{\left(b-a\right)^{2}+1}{\left(b-a-1\right)\left(b-a+1\right)}.
\end{eqnarray*}

.. math:: :nowrap:

\begin{eqnarray*}
    M\left(t\right) & = & \frac{1}{b-a}\sum_{k=a}^{b-1}e^{tk}\\
                    & = & \frac{e^{bt}-e^{at}}{\left(b-a\right)\left(e^{t}-1\right)}
\end{eqnarray*}

Implementation: scipy.stats.randint