What is the joint probability of X and Y on a unit interval of (0,1)?

[cse63146]

Homework Statement

Let U₁...U_n be independent and uniformly distributed over the unit interval (0,1). Let X be the minimum of U₁...U_n 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

Homework Equations

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?

 

[cse63146]

Anyone?

 

[HallsofIvy]

Homework Statement

Let U₁...U_n be independent and uniformly distributed over the unit interval (0,1). Let X be the minimum of U₁...U_n 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

Homework Equations

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?

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.

 

[cse63146]

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

 

[HallsofIvy]

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

 

[cse63146]

Any ideas?