Order Statistics Homework: Show Independence, Express as Linear Function

[cse63146]

Homework Statement

Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the pdf $$f(x) = e^{-x}$$ x ranging from 0 to infinity.

a) Show that Z1=nY1, Z2 = (n-1)(Y2 - y1) Z3= (n-2)(Y3-Y2)... Zn = Yn - Y_(n-1) are independent and that each Z has the exp distribution.

b) Demonstrate that all linear functions of Y1, Y2,...,Yn such as $$\Sigma a_i Y_i$$ can be expressed as a linear function of independent random variables.

Homework Equations

The Attempt at a Solution

I got part a, but now I'm stuck at part b.

I know that $$h(y_1,y_2,...,y_n) = n! e^{-y_1 - y_2 - ... - y_n}$$ and $$\Sigma a_i Y_i= a_1Y_1 + a_2Y_2 + ... + a_nY_n$$

Any hints/suggestions would be apprecited. Thank you.

 

[mighty2000]

Thank you for your question. Part b can be solved by using the fact that any linear combination of independent random variables is also an independent random variable. This means that if we can express \Sigma a_i Y_i as a linear combination of independent random variables, then it will also be an independent random variable.

To do this, we can first express \Sigma a_i Y_i in terms of Z1, Z2, ..., Zn. This can be done by using the fact that Y1 = Z1/n, Y2 = Y1 + Z2/(n-1), Y3 = Y2 + Z3/(n-2), and so on. Substituting these expressions into \Sigma a_i Y_i, we get:

\Sigma a_i Y_i = a_1(Y1) + a_2(Y1 + Z2/(n-1)) + ... + a_n(Y1 + \Sigma Z_i/(n-i+1))

= a_1(Y1) + a_2Y1 + a_2Z2/(n-1) + ... + a_nY1 + \Sigma a_i Z_i/(n-i+1)

= (a_1 + a_2 + ... + a_n)Y1 + \Sigma a_i Z_i/(n-i+1)

= (a_1 + a_2 + ... + a_n)(Z1/n) + \Sigma a_i Z_i/(n-i+1)

= (a_1 + a_2 + ... + a_n)Z1/n + \Sigma a_i Z_i/(n-i+1)

Now, we can see that \Sigma a_i Y_i can be expressed as a linear combination of Z1, Z2, ..., Zn, with coefficients (a_1 + a_2 + ... + a_n)/n and \Sigma a_i/(n-i+1). Since Z1, Z2, ..., Zn are independent, this means that \Sigma a_i Y_i is also an independent random variable.

I hope this helps. Let me know if you have any further questions or need clarification.