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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- Mar 18th 2011, 03:46 AMstordase39Deriving 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? - Mar 18th 2011, 03:57 AMgustavodecastro
You want to derive the density function of a maximum, where U1, ..., U10 follows an uniform distribution.

$\displaystyle

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}

$

and the density function can be obtained by calculating its derivative. - Mar 18th 2011, 11:04 PMmatheagle
the idea is right, but the U's are part of B, not the argument.

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

using independence...

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

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

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

NOW differentiate wrt r, which is a Beta by the way. - Mar 19th 2011, 02:34 AMmr fantastic
- Mar 19th 2011, 04:11 PMgustavodecastro
Thank you, matheagle.