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Math Help - Deriving denisty function for a given random variable

  1. #1
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    Deriving denisty function for a given random variable

    Lets say I have 10 random numbers U(0,1). I have a variable called B that is defined as B=max{U1,U2,...U10}

    I want help to prove that the density function of B is given by;

    f_B(r) = 10*r^9 where r is in the interval [0,1]

    How can I show this?
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  2. #2
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    You want to derive the density function of a maximum, where U1, ..., U10 follows an uniform distribution.

    <br />
P(B \leq U_1, \dots, U_{10}) = P(B \leq U_1) \dots P(B \leq U_{10}) = P(B \leq U_1)^{10}<br />

    and the density function can be obtained by calculating its derivative.
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  3. #3
    MHF Contributor matheagle's Avatar
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    the idea is right, but the U's are part of B, not the argument.

    F_B(r)=P(B\le r)=P(U_1\le r, U_2\le r,\cdots , U_{10}\le r)

    using independence...

    =P(U_1\le r)P(U_2\le r)\cdots P(U_{10}\le r)

    =\left(F_{U_1}(r)\right)^{10}=r^{10}

    Since the cummulative distribution of a U(0,1) is F_{U_1}(r)=r when 0<r<1.

    NOW differentiate wrt r, which is a Beta by the way.
    Last edited by matheagle; March 19th 2011 at 11:49 PM.
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  4. #4
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    Quote Originally Posted by stordase39 View Post
    Lets say I have 10 random numbers U(0,1). I have a variable called B that is defined as B=max{U1,U2,...U10}

    I want help to prove that the density function of B is given by;

    f_B(r) = 10*r^9 where r is in the interval [0,1]

    How can I show this?
    Do some research/background reading on Order Statistics.
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  5. #5
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    Thank you, matheagle.
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