Results 1 to 7 of 7

Math Help - pdf of area and circumference of a circle

  1. #1
    Newbie
    Joined
    May 2010
    Posts
    11

    pdf of area and circumference of a circle

    Suppose that the radius X of a circle is a random variable having the following p.d.f.:
    f(x)={ (1/8)(3x=1) for 0<x<2
    0 otherwise
    Determine the p.d.f. of the area of the circle and the circumference of the circle.

    can anyone help me with this one? do i just plug f(x) into the equations for area and circumference? how does that apply to the pdf of area and circumference???
    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 uva123 View Post
    Suppose that the radius X of a circle is a random variable having the following p.d.f.:
    f(x)={ (1/8)(3x=1) for 0<x<2
    0 otherwise
    Determine the p.d.f. of the area of the circle and the circumference of the circle.

    can anyone help me with this one? do i just plug f(x) into the equations for area and circumference? how does that apply to the pdf of area and circumference???
    You need to find the pdf of the random variables:

    1. A = \pi X^2

    2. C = 2 \pi X

    where it's known that the pdf of X is f(x) = \frac{3x + 1}{8} for 0 < x < 2 and 0 otherwise.

    You do NOT make a direct substitution. One way of doing it is to first find the cdf of each and then differentiate. What have you been taught about transforming random variables?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2010
    Posts
    11
    i havent learned about cpf's yet. i i cant figure our how to find the pdf of the area and circumference. that is where i am getting stuck.
    thanks for you help!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by uva123 View Post
    i havent learned about cpf's yet. i i cant figure our how to find the pdf of the area and circumference. that is where i am getting stuck.
    thanks for you help!
    The cdf of A is given by:

    G(a) = \Pr(A < a) = \Pr(\pi X^2 < a) = \Pr\left( - \frac{\sqrt{a}}{\sqrt{\pi}} < X < \frac{\sqrt{a}}{\sqrt{\pi}}\right) = \int_{0}^{\frac{\sqrt{a}}{\sqrt{\pi}}} f(x) \, dx = ....

    for 0 < a < 4 \pi

    and so the pdf of A is given by g(a) = \frac{dG}{da} = ....

    The details are left for you. The pdf of C can be found in a similar way.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    May 2010
    Posts
    11
    how did you find ? is it the same for c?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    Quote Originally Posted by uva123 View Post
    Suppose that the radius X of a circle is a random variable having the following p.d.f.:
    f(x)={ (1/8)(3x=1) for 0<x<2
    0 otherwise
    Determine the p.d.f. of the area of the circle and the circumference of the circle.

    can anyone help me with this one? do i just plug f(x) into the equations for area and circumference? how does that apply to the pdf of area and circumference???

    is that = a +?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by uva123 View Post
    how did you find ? is it the same for c?
    What restriction does the support of X place on the support of A? On C?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Area and Circumference of the circle relation
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 18th 2011, 03:16 PM
  2. Circumference and Area of circle
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: October 18th 2011, 11:03 AM
  3. Replies: 4
    Last Post: February 10th 2011, 08:14 AM
  4. Circle area and circumference
    Posted in the Geometry Forum
    Replies: 2
    Last Post: August 30th 2009, 03:19 PM
  5. Replies: 3
    Last Post: April 3rd 2009, 07:15 PM

Search Tags


/mathhelpforum @mathhelpforum