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.

Printable View

- August 2nd 2009, 11:16 AMmorganforDistribution problem
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.

- August 2nd 2009, 06:18 PMmatheagle
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.

- August 3rd 2009, 12:08 AMMoo
Hello,

Here is the good ol'trick of the "cumulative" probability ! (better than**bald**eagle's method :D)

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

So finally,