# Determine P(X > x, Y < y)

1. Jul 27, 2009

### cse63146

1. The problem statement, all variables and given/known data

Let U1...Un be independant and uniformly distributed over the unit interval (0,1). Let X be the minimum of U1...Un and Y be the maximum

a) Determine P(X > x, Y < y). Consider the following cases: 1) 0< x < y < 1 2) 0 < y 1, x < 0
3) 0 < x < 1, y > 1 4) x < 0, y > 1. 5) all remaining possibilites
b) Determine the joint CDF of X and Y
c) using b), determine a joint density funtion of X and Y

2. Relevant equations

3. The attempt at a solution

for a), is the only possible case that can occur is (1)? since it's on the interval (0,1) so X/Y cannot be smaller than 0, and cannot be bigger than 1? And Y also has to be greater than X, since X is the minimum and Y the maximum.

$$\int^1_0 \int^Y_0 dx dy$$

it doesn't seem right. Any hints?

2. Jul 30, 2009

### cse63146

Anyone?

3. Jul 30, 2009

### HallsofIvy

There is something wrong with your problem. You say X and Y are independent. If so, then there is no reason to "Consider the following cases". The probability that X> x is 1- x. The probability that Y< y is y. The probability that X> x and Y<y is (1- x)y.

And, the CDF of the joint probability if just P(X>x and Y>y)= xy.

4. Jul 30, 2009

### cse63146

if U1,U2,...,Un are independant, then so are X & Y?

5. Jul 30, 2009

### HallsofIvy

Oh, blast! I misread the problem. I thought you were saying X and Y were independent.

6. Jul 30, 2009

Any ideas?