Checking the Independence.

Jesmin

New Member
Let $$X_i\sim N(\mu,\sigma^2)$$ ; where $$[i=1,2,\ldots,n]$$
$$Z_i\sim N(0,1)$$ ; where $$[i=1,2,\ldots,n]$$

Proof that $$\frac{(\bar X-\mu)}{\sigma}$$ and $$\sum_{i=1}^n\frac{(X_i-\bar X)^2}{\sigma^2}$$ are independent, which implies $$\bar X$$ and $$\sum_{i=1}^n(X_i-\bar X)^2$$ are independent.

If i show that $$\bar X$$ and $$(X_i-\bar X)$$ are independent and draw conclusion that since $$\bar Z$$ is only the function of $$\bar X$$ and $$(Z_i-\bar Z)^2$$ is only the function of $$(X_i-\bar X)$$, so $$\bar Z$$ and $$\sum(Z_i-\bar Z)^2$$ are independent. Is it suffices, ie, is the procedure to check the independence correct?

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