Sum of squared uniform random variables

  • Thread starter mjkato
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  • #1
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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?
 

Answers and Replies

  • #2
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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
 

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