My probability book defines a discrete uniform random variable as a variable X such that P(X=x) = \frac{1}{b-a+1}, for all x=a,a+1...b. My doubt is, in a discrete uniform distribution must the numbers that the random variable may take on always be intenger and sequential? And if so, why? I mean, could there be a uniform distribution of the following form?
$P(X=x) = \begin{cases}\frac{1}{4} &,& \text{if }x \in\{ 3,5,6.4,9\} \\
0 &, & \text{otherwise} \end{cases}$
Athough it does make sense to me (since there are 4 possible outcomes with equal probability, so each individual probability must = 1/4), it doesn't seem to fit that definition: $\frac{1}{b-a+1} = \frac{1}{9-3+1}$ is different from $\frac{1}{4}$.
What am I getting wrong? Thanks in advance.