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Let $X$, $Y$ and $Z$ be random variables, with $Z$ independent from $X$ and from $Y$ (i.e. $P(X≤x,Z≤z) = P(X≤x)P(Z≤z)$ and $P(Y≤y,Z≤z) = P(Y≤y)P(Z≤z)$).

Is it true that a function $f(X,Y)$ is independent from $Z$ (i.e. $P(f(X,Y) ≤ a, Z ≤ z) = P(f(X,Y) ≤ a)P(Z ≤ z)$)?

I'm asking this question because a proof I'm studying derived the independence of sample variance ($s^2$) and sample mean ($m$) from the fact that differences $X_i - m$ were independent from $m$, and $s^2$ was a function of $X_1 - m,...,X_n - m$.

EDIT: I forgot to add that the sample is supposed to be random normal, but that wasn't the main point of my question. I understand all the proof, save that step I mentioned.

Jean
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  • The proof you are referring to regarding independence of sample mean and sample variance in normal distribution is based on the fact that if two random variables $X$ and $Y$ are independent, then $f(X)$ and $Y$ are also independent for any (measurable) function $f$: https://math.stackexchange.com/q/8742/321264. – StubbornAtom Oct 04 '20 at 10:34
  • Yes, I forgot to add that sample is supposed to be random normal. However my problem is that $X'=f(X_1,...,X_n)$ with $Y$ ind. $X_1$,..., $Y$ ind. $X_n$, and to me is not obvious that this implies $Y$ ind. $X'$. – Jean Oct 04 '20 at 11:00
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    In my last comment, take $X=(X_1-\overline X,\ldots,X_n-\overline X)$ and $Y=\overline X$. Since $X_i-\overline X$ is independent of $\overline X$ for every $i$, the random vector $X$ is independent of $Y$. And therefore $s^2=f(X)$ is independent of $Y$. – StubbornAtom Oct 04 '20 at 11:07

2 Answers2

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In two independent tosses of a fair coin the events $A,B,C$ of getting heads in the first toss, that of getting tails in the second and the event HH+TT are pairwise independent but not jointly independent. Take $X=I_A,Y=I_B, Z=I_C$ and $f(x,y)=x+y$ for a counter-example. You can see that $P(f(X,Y)=0, Z=1) \neq P(f(X,Y)=0) P(Z=1)$.

Kavi Rama Murthy
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Reading the purpose of your intent,

I'm asking this question because a proof I'm studying derived the independence of sample variance (s^2) and sample mean (m)

$\overline{X}_n$ and $S^2$ are not independent in general. They are independent in a gaussian model and this can be proved observing that

  • $\overline{X}_n$ is a complete and sufficient statistic for $\mu$ (Canonical Statistic of an Exponential Family Model)

  • $\frac{(n-1)S^2}{\sigma^2}\sim \chi_{(n-1)}^2$ thus $S^2$ is ancillary for $\mu$

Thus using Basu's Theorem, $\overline{X}_n$ and $S^2$ are independent

tommik
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