The two continuous random variables, and , are independent and have the same pdf,
clearly , we can say that and , are IID random variables.
Define the tow variables U and V as follows.
Now , the JACOBIAN of U and V is
1.
Now , the joint pdf of U and V ,
2.
Therefore , the marginal PDF of U is ,
where,
[consider u=x]
3, Therefore , the marginal PDF of V is ,
where,
[consider v=x]
4.
Yes, U and V are independent as the joint PDF of U and V is equal to the multiplication of the marginal PDF of U and V.
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