Discrete Uniform Distribution

#Math

$X\sim \text{Uniform}({1,2,\ldots,m})$

• A random variable is uniformly distributed if the probability of each $x\in S$ is equal. For the probabilities to be equal and add up to one, and since there are $m$ of $x\in S$, $p_X(x_i)=\frac{1}{m}$
• $p_X(x)=\frac{1}{m},~x\in{1,2,\ldots,m}$
• $F_X(x) =$
• $M_X(t) =$
• $\mathbb{E}[X]=\frac{m+1}{2}$
• $\text{var}(X) =\frac{m^{2}-1}{12}$
• An example of a random variable would be $X=$ the number of pips on a die face after a roll. There are 6 outcomes because $m=|{1,2,3,4,5,6}|=6$, meaning $p_X(x_i)=\frac{1}{m}=\frac{1}{6}$