Suppose Y_1,..., Y_n is a random sample where the density of each random variable Y_i is f(y) = 2*x^2*y^(-3), y >= 0 for some parameter x > 1. Find the first order statistic, Y_(1).

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- Mar 21st 2011, 02:52 PMjmyers12345Find the first order statistic
Suppose Y_1,..., Y_n is a random sample where the density of each random variable Y_i is f(y) = 2*x^2*y^(-3), y >= 0 for some parameter x > 1. Find the first order statistic, Y_(1).

- Mar 21st 2011, 04:37 PMmatheagle
use the complement, let M be the min

now if the min exceeds the value, y, they all do...

using independence

now differentiate and you have

finally plug in your density/distribution. - Mar 21st 2011, 04:51 PMjmyers12345
Thanks, that's what I had but I'm still getting the wrong answer. When computing the pdf and I'm integrating the cdf given what should my limits of integration be?

- Mar 21st 2011, 04:56 PMmatheagle
show your work and we can better answer your question

y goes from zero to infinity, x is a fixed constant.

NEVERMIND

this is NOT a valid density

MY guess is that y is bounded below by x, clearly it is not bounded below by 0. - Mar 21st 2011, 05:53 PMjmyers12345
Ok, so when I integrate (2*x^2*y^(-3)) with respect to y I get -(x^2/y^2) and plugging in the bounds of x to infinity, I get 0 - (-1) = 1. But this can't be right because then the entire pdf for the order statistic ends up equaling 0. Would I just use the indefinite integral of the pdf -(x^2/y^2) as my cdf and plug that in for the cdf found in the equation for the first order statistic that you listed above?

- Mar 21st 2011, 06:02 PMmatheagle
IT is right

ONE means it is a valid density

that's how I figured out what your density must be - Mar 21st 2011, 06:03 PMmatheagle

x is not a variable, it's just a fixed constant.

BUT you should look up the problem again, having y bounded below by zero, seems off.

must be one. - Mar 21st 2011, 06:05 PMjmyers12345
Thanks for all the help, I got it

- Mar 21st 2011, 06:09 PMmatheagle
but is that the correct density, where y>x?

- Mar 21st 2011, 06:13 PMjmyers12345