Sum of squared uniform random variables

[mjkato]

Homework Statement

If X and Y are independent uniformly distributed random variables between 0 and 1, what is the probability that X^2+Y^2 is less than or equal to one.

Homework Equations

P(Z<1) = P(X^2+Y^2<1)

For z between 0 and 1, P(X^2<z) = P(X < √z) = √z

The Attempt at a Solution

I'm a tad lost- I assume what you'd be looking for is the probability that Y^2 is less than or equal 1-X^2, or rather that Y is less or equal to √(1-X^2). Is that all there is to it?

 

[mtayab1994]

Homework Statement

If X and Y are independent uniformly distributed random variables between 0 and 1, what is the probability that X^2+Y^2 is less than or equal to one.

Homework Equations

P(Z<1) = P(X^2+Y^2<1)

For z between 0 and 1, P(X^2<z) = P(X < √z) = √z

The Attempt at a Solution

I'm a tad lost- I assume what you'd be looking for is the probability that Y^2 is less than or equal 1-X^2, or rather that Y is less or equal to √(1-X^2). Is that all there is to it?

You have to remember something really important about variables between 1 and zero. Here it is:

if xε[0,1[ then x^2<x and √x> x I think that should help you a bit