i have following problem,
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 the expectation, standard deviation and $P(X\le 0)$.
thinking about it, i believe that expectation is the value of X when $f(x)=0.997356$.
can you please help?
The distribution is symmetric wrt to $\mu$ leading to $F(\mu-x)+F(\mu+x)=1$ for every $x$.
Keeping this in mind the equality $F(-1)+F(7)=1$ enables you to find $\mu$.
Further $f(x)$ takes $\frac1{\sigma\sqrt{2\pi}}$ as maximum enabling you to find $\sigma$.
Knowing $\mu$ and $\sigma$ you know the distribution so can find $P(X\leq0)$.