Deriving denisty function for a given random variable

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• Mar 18th 2011, 03:46 AM
stordase39
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?
• Mar 18th 2011, 03:57 AM
gustavodecastro
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 PM
matheagle
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 AM
mr fantastic
Quote:

Originally Posted by stordase39
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.
• Mar 19th 2011, 04:11 PM
gustavodecastro
Thank you, matheagle.