Finding the Distribution of Minimum Random Variables in Uniform Distribution?

[roeb]

Homework Statement

Let X₁, X₂, ... X_n 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.

Homework Equations

The Attempt at a Solution

I can see that P(X_i = j) = 1/K (uniformly distributed), where i = 1, 2, 3, .. n and j = 1, 2, 3, .. k and Y = min (X₁, X₂, ..., X_n)

So using a 'trick' that my professor told us, {at least x} - {at least x + 1} = {exactly x} -- but shouldn't that be reversed to get "x" or are we assuming that x+1 < x?

My professor also said that (P(X_i>= x))ⁿ = ( (k - x + 1)/k )ⁿ but I'm having a hard time seeing where this is derived from. Can anyone explain it?

Ultimately, P( Y = x) = ( (k-x+1)^n - (k-x)^n ) / k^n

 

[statdad]

"{at least x} - {at least x + 1} = {exactly x}"

To visualize this pick a specific x for a concrete example: I choose x = 10

at least 10 = 10, 11, 12, 13, ...

at least 11 = 11, 12, 13,...

(at least 10) - (at least 11) = 10

"(P(Xi>= x))n = ( (k - x + 1)/k )n"

Suppose Xi >= x. Of the k total values there are k - x + 1 smaller than x, so
P(Xi < x ) = (k-x+1)/k. Raise both sides to the nth power.

 

[roeb]

Understood, thanks!