Results 1 to 5 of 5

Math Help - Uniform distribution on a circle

  1. #1
    Member
    Joined
    Sep 2008
    Posts
    79

    Uniform distribution on a circle

    Suppose that (X, Y ) is uniformly distributed on a circle with center at the origin and radius. What is P(3X + Y > 1)?
    Hint: Sketch the region over which you must integrate. Also, an antiderivative of sqrt(1-y^2) is .5y*sqrt(1-y^2) +.5*arcsin(y)


    First I found the pdf of (X,Y) by taking the double integral of a constant c from -sqrt(1-y^2) to sqrt(1-y^2) for dx and from -1 to 1 for dy, and then set it equal to 1 to find c = 1/pi. Then to find P(3X + Y > 1), I took the integral of 1/pi from (1/3)-(1/3)y to sqrt(1-y^2) for dx and from -4/5 to 1 for dy. This gave me an answer of 8.61, which can't possibly be right for a probability. Did I set this problem up incorrectly? Sorry if this explanation is hard to understand. I will clarify it if needed.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by uberbandgeek6 View Post
    Suppose that (X, Y ) is uniformly distributed on a circle with center at the origin and radius. What is P(3X + Y > 1)?
    Hint: Sketch the region over which you must integrate. Also, an antiderivative of sqrt(1-y^2) is .5y*sqrt(1-y^2) +.5*arcsin(y)


    First I found the pdf of (X,Y) by taking the double integral of a constant c from -sqrt(1-y^2) to sqrt(1-y^2) for dx and from -1 to 1 for dy, and then set it equal to 1 to find c = 1/pi. Then to find P(3X + Y > 1), I took the integral of 1/pi from (1/3)-(1/3)y to sqrt(1-y^2) for dx and from -4/5 to 1 for dy. This gave me an answer of 8.61, which can't possibly be right for a probability. Did I set this problem up incorrectly? Sorry if this explanation is hard to understand. I will clarify it if needed.
    I'd make the change of variable X = R \cos \theta and Y = R \sin \theta where R ~ U(0, 1) and \theta ~  U(0, 2\pi) and use the Change of Variable Theorem to get the joint pdf of X and Y.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Sep 2008
    Posts
    79
    That doesn't really seem necessary when I'm given the antiderivative of sqrt(1-y^2) in the problem. I think f(x,y) must be 1/pi because taking the integral of it from -sqrt(1-y^2) to sqrt(1-y^2) for dx and then from -1 to 1 for dy gives 1. Its just when I try to find P(3X + Y > 1) by taking the double integral of 1/pi from (1/3)-(1/3)y to sqrt(1-y^2) for dx and from -4/5 to 1 for dy that I get an answer > 1. Shouldn't this integral give me the answer I'm looking for?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    IF this is a uniform over any region, I would NOT use calculus.
    Just find the area of interest over the total area, i.e., use geometry.
    IF I used calc I would only obtain the probability, not a new density.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by uberbandgeek6 View Post
    Suppose that (X, Y ) is uniformly distributed on a circle with center at the origin and radius. What is P(3X + Y > 1)?
    Hint: Sketch the region over which you must integrate. Also, an antiderivative of sqrt(1-y^2) is .5y*sqrt(1-y^2) +.5*arcsin(y)
    Do what the hint is hinting at - draw a picture. Then as MathEagle says the required probability is the ratio of the area of the region of interest to that of the whole circle.

    (also it is a disk not a circle, a circle is the line around the edge).

    CB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Mixing a uniform distribution with a normal distribution
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: July 8th 2011, 09:27 AM
  2. Uniform distribution
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: February 23rd 2010, 11:36 PM
  3. Uniform Distribution of a Circle
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: November 23rd 2009, 12:45 PM
  4. Uniform Distribution
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: October 2nd 2009, 05:54 AM
  5. Non-Uniform Motion in a circle
    Posted in the Advanced Applied Math Forum
    Replies: 2
    Last Post: July 7th 2008, 05:06 AM

Search Tags


/mathhelpforum @mathhelpforum