$X$ follows normal distribution $\mathcal{N}(\mu , \sigma^2)$ with pdf $f$ and cdf $F$. If $\max_x f(x)= 0.997356$ and $F(-1)+F(7)=1$, determine $\mu, \sigma^2$ and $\mathbb{P}[X\le 0]$ .
I have no clue about this question and unable to interpret the given conditions. How can I relate the max pdf to find the mean. Even if I get to know the first part I can calculate the rest. Any help would be grateful.
HINT
Since $X \sim \mathcal{N}\left(\mu,\sigma^2\right)$, you know that $$ f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(\frac{-1}{2} \left(\frac{x-\mu}{\sigma} \right)^2 \right) $$ which is the bell curve, clearly reaching maximum at $x = \mu$. What is this maximum, and can you find $\sigma$ from its value?
Then use the fact that $(X-\mu)/\sigma \sim \mathcal{N}(0,1)$ and usual relationship of the std normal cdf to figure out $\mu$ from the second relation.