# rectangular distribution problem

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• Apr 6th 2008, 04:41 AM
BklynKid
rectangular distribution problem
Having another problem. Not sure how to even approach this problem. I figure I should be using rectangular distribution somehow.

Let X and Y denote the lengths of the two sides of a rectangle and let them possess independent rectangular distributions over the interval (0,1).
Calculate the probability that the length of a diagonal of this rectangle will be less than one.

Any help would be appreciated.
• Apr 6th 2008, 05:34 AM
CaptainBlack
Quote:

Originally Posted by BklynKid
Having another problem. Not sure how to even approach this problem. I figure I should be using rectangular distribution somehow.

Let X and Y denote the lengths of the two sides of a rectangle and let them possess independent rectangular distributions over the interval (0,1).
Calculate the probability that the length of a diagonal of this rectangle will be less than one.

Any help would be appreciated.

The square of the diagonal of the random rectangle is $\displaystyle d^2=x^2+y^2$, the probability that the diagonal is less than one is: $\displaystyle p(x^2+y^2<1)$, which as $\displaystyle x \sim U(0,1)$ and $\displaystyle y \sim U(0,1)$, is the probability that a random point in the square $\displaystyle [0,1]\times [0,1]$ lies inside the unit circle.

This probability is equal to the area of the unit circle inside the square, which is a quarter of the area or the unit circle or $\displaystyle \pi/4$

RonL
• Apr 6th 2008, 08:33 AM
BklynKid
Thanks for the swift reply. To be honest, I don't really understand the reason (I already knew the solution) but perhaps it'll come to me later.

Again, thank you.