Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Simple question about probability

  1. Oct 14, 2013 #1
    Hi All,
    I'm stucked computing this:
    I have two discrete random variables 1≤X≤N and 1≤Y≤N. How many pairs of (X,Y) satisfy X²+Y²≤N²

    I began by using a certain value for N and trying to search for patterns in the numbers that satisfy this constraint but I can't seem to get any meaningful pattern.

    Any ideas?
  2. jcsd
  3. Oct 14, 2013 #2


    User Avatar
    Science Advisor

    Try drawing it out. Draw the region 1<=X<=N and 1<=Y<=N in the X-Y plane. Then draw in the condition X^2 +Y^2 <= N^2. Does this help?
  4. Oct 15, 2013 #3
    At first I draw it in a paper with N=9. And that give me no clue. Today, with the purpose to show you the figure, I made it again, but in Mathematica. When I see the elementes that satisfy the condition in that matrix, the answer to N→∞ become so obvius. I attach the image.

    When N→∞ the amount of numbers that satisty X2+Y2≤N2 is Pi*N2/4, because X,Y>0. If X and Y be any real number equal to the restriction is an equation of a circle, treats including its inside.
    For small N, I still have no idea.

    Thanks, phyzguy.
  5. Oct 15, 2013 #4
    I forgot to post the images.

    Attached Files:

    • 1.PNG
      File size:
      16.8 KB
    • N500.PNG
      File size:
      4.3 KB
  6. Oct 15, 2013 #5


    User Avatar
    Science Advisor

    Since the title of the thread said it was a probability question, I assumed it was in the large N limit, since this is what a probability is. Are you supposed to calculate the number of pairs as a function of N. If so, I don't know how to write a general formula.
  7. Oct 15, 2013 #6


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    If the question is simply "how many integer pairs (X,Y) are there such that X2+Y2 <= N2", then that doesn't have a lot to do with probability. Unless your random variables X and Y have some specific distribution besides the uniform distribution that you neglected to tell us, this seems to be the question you're asking.
  8. Oct 15, 2013 #7
    I agree, my fault. The thing is i need to calculate the probability of that event, so I need to know how many pairs satisfy that condition. I already know that the probability of any pair to come up is 1/N^2, therefore I only need to know how many pairs like that can come up and multiply that number by 1/N^2. That's why I asked for the number of pairs.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook