Hi: I am reading an article that deals with the distribution function associated with the half-normal distribution. The author presents a formula for the c.d.f. as:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \left [ G\left ( x \right ) \right ]^{2r-2}=\left ( \frac{2}{\pi } \right )^{r-1}\left \{ \int_{0}^{x}\exp \left [ -\frac{1}{2} \left (w ^{2}\right )\right ]dw \right \}^{2r-2} [/tex]

Note thatr=2,3,...

The expression above is also equal to:

[tex] \left [ G\left ( x \right ) \right ]^{2r-2}=\left ( \frac{2}{\pi } \right )^{r-1}\left \{ \int_{0}^{x}\int_{0}^{x} \exp \left [- \frac{1}{2}\left ( x_{1}^{2}+x_{2}^{2} \right ) \right ]dx_{1}dx_{2}\right \}^{r-1} [/tex]

which I have no problem with. The author then states that the second equation above is also equal to:

[tex] \left [ G\left ( x \right ) \right ]^{2r-2}=\left ( \frac{4}{\pi } \right )^{r-1}\left \{ \int_{0}^{\frac{\pi }{4}}\left [ 1-\exp \left ( -\frac{1}{2}x^{2}\sec ^{2}\theta \right ) \right ]d\theta \right \}^{r-1} [/tex]

Can someone explain to me how the author gets from the second equation to the third (or last) equation above.

Thanks in advance.

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# Half-normal distribution

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