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Suppose $X$ ~ $N(3,5)$ and $Y$ ~ $N(-7,2)$ be independent.
Find constants to satisfy:

$\quad C_1(X+C_2)^2$ + $C_3(Y+C_4)^2$ ~ $\chi^2(C_5)$

I started with:
$\quad Z_1=\sqrt{C_1}X+\sqrt{C_1}C_2$ ~ $N(3\sqrt{C_1}+C_2\sqrt{C_1}, \;5C_1)$
$\quad Z_2=\sqrt{C_3}Y+\sqrt{C_3}C_4$ ~ $N(-7\sqrt{C_3}+C_4\sqrt{C_3}, \;2C_4)$

To satisfy $Z_1^2$ + $Z_2^2$ ~ $\chi^2(2)$ we need:
$\quad C_5=2$ and $Z_1, Z_2$~$N(0, 1)$

which gives:
$\quad C_1=\frac{1}{5}$
$\quad C_2=-3$
$\quad C_3=\frac{1}{2}$
$\quad C_4=7$

However, the solution is supposed to be:
$\quad C_1=\frac{1}{\sqrt{5}}$
$\quad C_2=-3$
$\quad C_3=\frac{1}{\sqrt{2}}$
$\quad C_4=7$

I'd like to know where I messed up the calculation?

user137481
  • 2,605

1 Answers1

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$\frac{X-3}{\sqrt{5}}$ and $\frac{Y+7}{\sqrt{2}}$ are independent standard normals. Thus, the sum of squares of these is $\chi^2(2)$.

This gives $C_2=-3$, $C_4 = 7$, $C_5=2$, $C_1=1/5$ and $C_3=1/2$.

Vaneet
  • 1,493
  • thanks. But I'm looking for constants that satisfy $Z_1^2 +Z_2^2$ ~ $\chi^2(2)$. – user137481 Jun 05 '16 at 21:56
  • Look at Theorem 2 in https://onlinecourses.science.psu.edu/stat414/node/171 . Sum of squares of independent Gaussians will be $\chi^2(2)$. That is what has been used. – Vaneet Jun 05 '16 at 21:59
  • ok, I now understand what you meant. But since $\left(\frac{X-3}{\sqrt{5}}\right)^2 + \left(\frac{Y+7}{\sqrt{2}}\right)^2=\frac{1}{5}\left(X-3\right)^2 + \frac{1}{2}\left(Y+7\right)^2$ we should $C_1=\frac{1}{5}$ but you have $C_1 = 5$. Is that a typo? – user137481 Jun 05 '16 at 23:53
  • Yes, $C1=1/5$, $C3=1/2$. – Vaneet Jun 06 '16 at 00:08