Why does the cumulative distribution function of the standard normal distribution have point symmetry in (0,0.5)?
The probability distribution function for the standard normal distribution, $\phi(z)=\frac{e^{-\frac{1}{2}z^2}}{\sqrt{2\pi}}$ is even, with line symmetry in the y-axis.
The cumulative distribution function of the standard normal distribution is $\Phi(z)=\int_{-\infty}^z\phi(t)dt$.
My textbook said that
Because $\phi(z)$ is even, $\Phi(z)$ has point symmetry in $(0,0.5)$.
Which makes sense to me by looking at their graphs (Since the PDF is the derivative of the CDF), but is there a more rigorous mathematical proof for why the CDF has point symmetry in (0,0.5)?
And more importantly, is there some general pattern/mathematical theory that I am missing here which I am supposed to know?
What I mean by this is, obviously, the primitive of $y=x^2$ (which is even) is $y=x^3$, which has point symmetry in the origin. But I was wondering if there is some greater mathematical theory that says something about 'if a function has line symmetry, its primitive will always have point symmetry' or something like that..
Thanks..
$\endgroup$1 Answer
$\begingroup$$$\Phi(x) = \int_{-\infty}^x \phi(z) dz \stackrel{z \to -z}{=} \int_{-x}^{+\infty} \phi(-z)dz = \int_{-x}^{+\infty} \phi(z)dz = 1 - \int_{-\infty}^{-x} \phi(z)dz = 1- \Phi(-x)$$
$\endgroup$ 0More in general
"Zoraya ter Beek, age 29, just died by assisted suicide in the Netherlands. She was physically healthy, but psychologically depressed. It's an abomination that an entire society would actively facilitate, even encourage, someone ending their own life because they had no hope. Th…"
