How to check if a distribution is different from uniform distribution using the values only? I have a discrete empirical distribution, where the values D1 = {1,2,3,7,9,15,16,17,18,19,20} occurs with some different frequencies. The frequencies are not of interest to me. I am interested in how many numbers are there in the distribution.
Now, if we generate 200 random numbers in the range 1 to 20, most of the times we will have a set, which consists of all the numbers from 1 to 20 at least once. Sometimes though there will be 19 numbers, 18 numbers and so on.  How do I say with a certain confidence that D1 is significantly different from the random distribution using this length of the set of the unique values?
 A: This is an occupancy problem, similar to a coupon collector or birthday problem 
If you have $d$ possible values/types of coupons/days in the year and $t$ independent uniform selections then the probability that $x$ different values have been selected is $$\frac{S_2(t,x)\,  d!}{d^t\, (d-x)!}$$ where $S_2(t,x)$ is a Stirling number of the second kind
With $t=200$ and $d=20$, this gives the probabilities for large values of $x$ of about:
 x       prob
20    0.9992991 
19    0.0007008
18    0.0000001

with even smaller probabilities for smaller values of $x$.  So the probability of seeing $19$ or fewer distinct values is less than $0.1\%$, and you can be more than $99.9\%$ confident of seeing all $20$ distinct values  if the null hypothesis of independent uniform selections is correct
As a check, the expected number of distinct values is $d \left(1- \left(1- \frac1{d}\right)^{t}\right)$ and with $t=200$ and $d=20$ this gives $20 \left(1- \left(1- \frac1{20}\right)^{200}\right) \approx 19.99929895$. Indeed, you can use this as a quick way of saying that the probability of observing strictly fewer than $20$ distinct values is less than $20$ minus this expectation, which is about $0.00070105$, though this shortcut only really works when $t \gg d \log_e(d)$ 
