Let X1, X2,...,Xn be n mutually independent random variables, each of which is uniformly distributed on the integers from 1 to k. Let Y denote the minimum of the Xi's. Find the distribution of Y.
The continuous case is easier, but the same logic works here.
Then it gets messy.
Here you have to think about all the possible strings of length n of numbers 1 through k.
All sequences are equally likely.
To have exactly one 2 (and no 1's) that's
To have exactly two 2's (and no 1's) that's
Then exactly three 2's....
The sum of all of these will give you P(Y=2).
The last is easy.
Here is the good ol'trick of the "cumulative" probability ! (better than baldeagle's method )
For any j, integer between 1 and k :
Similarly, for any j, integer between 1 and n-1, we have :
If j=k, this probability equals 0. So this formula works for any j, integer between 1 and k.
Since Y has integer values, we can easily see that