Let X1 and X2 be a random sample of size 2 from a distribution with probability density function f(x) = exp(-x) , 0 <x < infinity. and f(x)= 0 elsewhere.
Let X1 and X2 be a random sample of size 2 from a distribution with probability density function f(x) = exp(-x) , 0 <x < infinity. and f(x)= 0 elsewhere.
Now, let Y1= X1+ X2.
Y2 = X1/(X1+X2).
Prove that Y2 and Y2 are independent.
The joint pdf of and is .
and . Therefore .
Therefore the joint pdf of and is
Now show that can be written as a product of the form .
Aside: If and are independent then . The converse is NOT true.
, as expected.
Note: .
Therefore .
Last edited by mr fantastic; February 9th 2009 at 01:54 AM.
Let X1 and X2 be a random sample of size 2 from a distribution with probability density function f(x) = exp(-x) , 0 <x < infinity. and f(x)= 0 elsewhere.
Now, let Y1= X1+ X2.
Y2 = X1/(X1+X2).
Prove that Y2 and Y2 are independent.
Anyone wishing to contribute to this thread can pm me. In light of another thread being completely vandalised by edit-deletes, I'm closing this thread.