# What is the difference between two discrete uniform distribution with the same range but different number of categories?

Two random number generators with uniform distributions having min, max as (0,8) The first generates all integers between 0 and 8 uniformly. But the second generates only [0,2,4,6,8] uniformly.

What is the difference between the second generator and the first? Smaller range, but scaled up? A discrete vs. continuous distribution? Higher variance? How would one model the second distribution?

• Could you explain what you mean by "model" a distribution? – whuber Apr 25 '13 at 0:13
• Wouldn't the second distribution be just a multinomial distribution? – Robert Smith Apr 25 '13 at 4:14

They are both discrete uniform distributions. That is they both have constant probabilities for a finite number of values ($P(X=c) =1/k$ where $k$ is the number of values the variable can take and $c$ is a possible value of $X$.) They both have the same range and mean.
They have a different $k$ (i.e., number of values the variable can take).
A common representation of discrete uniform variables is $[1, 2, ..., n]$. Thus, $[0,1,..., 8]$ is just $[1,2,...,9] - 1$, and $[0, 2, ..., 8]$ is just $2 \times ([1,2,3,4,5] - 1)$
$[0,2,4,6,8]$ has larger variance. The variance of a $[1,2,3,...,n]$ uniform variable is $\frac{n^2 - 1}{12}$. Thus, the variance of $[0,1,...,8]$ is $\frac{9^2 - 1}{12} \approx 6.66$. The variance of $[0,2,4,6,8]$ is just $\textrm{Var}(2(X - 1)))$ where $X$ is $[1,2,3,4,5]$ which is $2^2 \times \textrm{Var}(X) = 4 \times \frac{5^2 - 1}{12} = 4 \times 2 = 8$.