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?
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?
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.
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.