Hello,
Hey... I think you have serious problems with the definitions... f is the pdf, the random variable does not equal
Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the pdf x ranging from 0 to infinity.
Demonstrate that all linear functions of Y1, Y2,...,Yn such as can be expressed as a linear function of independent random variables.
so:
That can't be right though....
are the order statistics of an Exponential(1) random variable; so are n independent (exponentially distributed) random variables.
A proof of this fact can be found in Feller, "An Introduction to Probability Theory and Its Applications, Volume II"; or you may be familiar with the statement that if arrival times are exponentially distributed then the inter-arrival times are independent and exponentially distributed.
Then
etc.,
so the Y's are linear functions of the W's. Hence any linear function of the Y's can be re-written as a linear function of the W's.
God, I'm such an idiot. There was another part to this question where it asks me to prove that Z1 = nY1 Z2 = (n-1)(Y2 - Y1) Z3 = (n-2)(Y3-Y2),...,Zn = Yn - Y(n-1)
so:
Can't believe I didn't realise this sooner. Thanks a lot.
Last question: is the nth term correct?