A stochastic variable X is said to follow the two-parameter Birnbaum-Saunders distribution, we write $X \in BS(\alpha,\beta) $ if its distribution function is
$$ F_X(x) = \begin{cases} \phi(\frac{1}{\alpha}(\sqrt{\frac{x}{\beta}}-\sqrt{\frac{\beta}{x}}) \qquad x > 0\\ 0 \qquad \text{else} \end{cases}$$
where $ \phi $ is the cumulative distribution function of $N(0,1)$.
Im a little confused with a distribution inside another distribution, is this correct?
Let $\gamma = \frac{1}{\alpha}(\sqrt{\frac{x}{\beta}}-\sqrt{\frac{\beta}{x}}). $
$$ F_X(x) = \int_{-\infty}^x (\int_{-\infty}^{\gamma}\frac{1}{\sqrt{2 \pi}}e^{-z^2/2}dz)d\gamma $$
for x > 0.
What's confusing me is that $ \gamma = \gamma(x).$