- #1
WMDhamnekar
MHB
- 377
- 28
Let X and Y be independent normal random variables each having parameters $\mu$ and $\sigma^2$. I want to show that X+Y is independet of X-Y without using Jacobian transformation.
Hint given by author:- Find their joint moment generating functions.
Answer: Now Joint MGf of $X+Y={e^{\mu}}^{2t}+\sigma^2t^2$ and of $X-Y=1$. So, joint MGF of $X+Y+X-Y$ is $e^{2\mu t}+ \sigma^2 t^2$. This indicates they are independent. Is their any other method in advanced calculus?
Hint given by author:- Find their joint moment generating functions.
Answer: Now Joint MGf of $X+Y={e^{\mu}}^{2t}+\sigma^2t^2$ and of $X-Y=1$. So, joint MGF of $X+Y+X-Y$ is $e^{2\mu t}+ \sigma^2 t^2$. This indicates they are independent. Is their any other method in advanced calculus?
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