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$$X_1, ...,X_n\text{~}N(\mu_x ,\sigma^2_x)^{i.i.d.} ~~~~~~~Y_1,...,Y_n\text{~}N(\mu_y ,\sigma^2_y)^{i.i.d.} $$

$$W=\frac {\sum_{i=1}^{n}(X_i-\mu_x)^2}{\sigma^2_{x}} + \frac {\sum_{i=1}^{n}(Y_i-\mu_y)^2}{\sigma^2_{y}}$$ $$Derive~~the~~distribution~~of~~W$$

I tried playing around with this but could not make any progress any help would be greatly appreciated.

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  • $\begingroup$ Can you identify the distribution of $\frac{X_i-\mu_x}{\sigma_x}$? How about $\left(\frac{X_i-\mu_x}{\sigma_x}\right)^2$? $\endgroup$ – Brent Kerby Feb 26 '15 at 2:03
  • $\begingroup$ If $X \sim N(\mu,1)$ then $X-\mu \sim N(0,1)$; if $X \sim N(0,1)$ then $\sigma X \sim N(0,\sigma^2)$; if $X \sim N(0,1)$ then $X^2 \sim \chi^2_{(1)}$; if $Y_i \sim \chi^2_{(1)}$ and $Y_i$ are independent, then $\sum_{i=1}^k Y_i \sim \chi^2_{(k)}$. $\endgroup$ – snar Feb 26 '15 at 2:29

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