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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?
    Thanks.
     
  2. jcsd
  3. Oct 14, 2013 #2

    phyzguy

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    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.
     

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  6. Oct 15, 2013 #5

    phyzguy

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    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

    Office_Shredder

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    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.
     
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