Let say we divide the interval $[0,1]$ in 10 equally large parts and we generate 100 different uniformly distributed numbers from the interval. Now let $X_1$ be the amount of numbers generated that is in the interval $[0,\frac{1}{10})$, the probability space of $X_1$ is $\{1,2,...,100\}$. Then (I hope) this $X_1$ is $Bin(100, 0.1)$.
Now here comes the question. Does the other $X_i, \ 1 < i \leq 10$ have the same distribution? Intuitively this would be, since all the intervals have the same length, but these are not independant since if $X_1=100$ then the rest of the $X_i$ must be equal to zero. Can we say anything about the other $X_i$?
Thanks in advance
The vector of counts $(X_1,\ldots,X_{10})$ has a multinomial distribution; each component of which has the same $n=100, p=1/10$ binomial distribution, and, as the OP indicates, the components are not independent. If one is interested in testing the uniformity of ones random numbers, one can use the evidence of all the $X_i$; the usual way to do this is with the chi-squared test.