My textbook shows the following step..
$$\large{F_{T}(t) = \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi}}e^{\frac{-y^{2}}{2}} \left(\int_{-\infty}^{t|y|} \frac{1}{\sqrt{2 \pi}}e^{\frac{-x^{2}}{2}} dx \right) dy}$$
$$\large{=\int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi}}e^{\frac{-y^{2}}{2}} \phi(t|y|) dy}$$
What is the meaning of $\phi$, and what 'happened' in this step? I understand how to take integrals generally but I have never seen this, and I wasn't able to find an explanation online (I'm not quite sure what to search?).
$\endgroup$ 11 Answer
$\begingroup$$\Phi(x)$ is simply shorthand for the cumulative normal density function. It is the integral of the Gaussian (normal) density from $-\infty$ to $x$
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