Uniform Distribution
Uniform distribution arises when considering that all possible values within a range are equiprobable. Therefore, the probability that a random variable will take values on a subinterval is proportional to the length of the same
We say that a random variable has a Uniform distribution on an interval of finite [a, b], and we denote it as \xi \approx U(a, b) if its density function is:
f(x)=\begin{cases} \frac{1}{b -a},\text{ if }a\leq x\leq b \\ 0,\text{ in the rest} \end{cases}
Its probability function is:
P\{\xi < k \} = \begin{cases} 0, \text{ if }x < a \\ \frac{x-a}{b-a}, \text{ if }a\leq x\leq b \\ 1, \text{ if } x > b \end{cases}
E(\xi) = \frac{a+b}{2}
\sigma^2(\xi) = \frac{(a+b)^2}{12}
\sigma(\xi) = (a+b)\cdot +\sqrt{\frac{1}{12}}