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Math Help - Density of two discrete uniform random variables

  1. #1
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    Density of two discrete uniform random variables

    Let X and Y be independent random variables having the uniform density on {0, 1,..,N}. How do you find the density of min(X,Y)?

    Here's what I did.
    First i wrote out
    P(min(X,Y) >= z) = P(X >= z, Y >= z)
    = P(X >=z)*P(Y>=z)
    = (N+1 - z + 1)/(N+1) * (N+1 - z + 1)/(N+1)
    = (N+1-z+1)^2/(N+1)^2

    I'm not whether that is the correct way to start. And I don't know what to do after that. The solution provided by the book is [2(N-z)+1]/(N+1)^2.

    Can someone explain how to solve this problem?
    Thanks!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by feiyingx View Post
    Let X and Y be independent random variables having the uniform density on {0, 1,..,N}. How do you find the density of min(X,Y)?

    Here's what I did.
    First i wrote out
    P(min(X,Y) >= z) = P(X >= z, Y >= z)
    = P(X >=z)*P(Y>=z)
    = (N+1 - z + 1)/(N+1) * (N+1 - z + 1)/(N+1)
    = (N+1-z+1)^2/(N+1)^2

    I'm not whether that is the correct way to start. And I don't know what to do after that. The solution provided by the book is [2(N-z)+1]/(N+1)^2.

    Can someone explain how to solve this problem?
    Thanks!
    You can't have a density on a discrete sample space, you need a probability
    mass function.

    RonL
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  3. #3
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    How can we compute the mass function of min(X,Y)?
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  4. #4
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    Quote Originally Posted by feiyingx View Post
    How can we compute the mass function of min(X,Y)?
    p(x=c)=1/N, p(y=c)=1/N, p(x>c)=1-c/N, p(y>c)=1-c/N

    p(min(x,y)=c) = p(x=c)*p(y>c) + p(y=c)*p(x>c) + p(x=c)p(y=c)

    RonL
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    thanks
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