Is it true that
If $a$ is random vector independent of $x$ for which $P(a'\Sigma a=0)=0$, then $$f\sim N(0,I)$$ and $f$ is independent of $a$ ?
Here $x\sim N_p(\mu,\Sigma)$ and $f={a'(x-\mu)\over\sqrt{a'\Sigma a}}$
$f\sim N(0, I)$
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random-variables
normal-distribution
independence
1 Answers
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Use the fact that the distribution of $f$ is $N(0,1)$ whatever may be the value of $a$. This shows $f$ is distributed independent of $a$.