I don't understand how we can infer symmetry if CDFs sum to 1. Here is the problem and its solution I don't understand.
X follows a normal distribution $ N(\mu, \sigma) $ whose cdf satisfies F(-1) + F(7) =1 Determine $ \mu $
Solution: As F(-1) + F(-7) = 1, -1 and 7 are symmetric with respect to $ \mu $, hence $ \mu $ = 3
How we can conclude the symmetry here? How do we know these 2 points have the same distance to the mean of the distribution?
You can standardize the random variable: $Z=\frac{X-\mu}{\sigma}$. Then $\Phi(z)$ denotes cdf of the standard normal variable. So we start with $$\Phi\left(\frac{-1-\mu}{\sigma}\right)+\Phi\left(\frac{7-\mu}{\sigma}\right)=1$$
The random variable Z is symmetric around the mean $\mu=0$. That means $\Phi(-z)=1-\Phi(z)$. So the eqution above becomes
$$1-\Phi\left(\frac{1+\mu}{\sigma}\right)+\Phi\left(\frac{7-\mu}{\sigma}\right)=1$$
$$\Phi\left(\frac{1+\mu}{\sigma}\right)=\Phi\left(\frac{7-\mu}{\sigma}\right)$$
Finally compare the arguments of the cdf.