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WMDhamnekar

MHB

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Let X and Y be independent normal random variables each having parameters $\mu$ and $\sigma^2$. I want to show that

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?

**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?

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