Hello, everyone.

Sorry, my LaTeX is working "part-time"

Let

X1 ~ U(0,1)

X2 ~ U(0,1)

X1 and X2 are independent.

Now, consider

x1 = 0.05

x2 = 0.05

Let:

Z = $\displaystyle \left [\chi^2_3 \right ]^{-1}\left ( 1-x_1 \right )$

Y = $\displaystyle \left [\chi^2_1 \right ]^{-1}\left ( 1-x_2 \right )$

W = Z+Y

where

W ~ $\displaystyle \chi^2_4 $

In this case,

W = 7.814+3.841 = 11.656

Prob [ W> $\displaystyle \chi^2_4 $] = 0.00209975 for an alpha level = 5%.

So, my question is:

Is there a way to create

Z = $\displaystyle \left [\chi^2_n \right ]^{-1}\left ( 1-x_1 \right )$

Y = $\displaystyle \left [\chi^2_m \right ]^{-1}\left ( 1-x_2 \right )$

where n and m are large numbers (say, n>100000) - so that

W ~ $\displaystyle \chi^2_{m+n} $ and

Prob [ W> $\displaystyle \chi^2{m+n} $] is still equal (or approximately equal ) to 0.00209975 for an alpha level = 5%?