# find mean and standard deviation of normal distribution from pdf and CDF

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 $$f(x)=0.997356$$.

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 $$P(X\leq0)$$.
• I was able to find mean as 3, but didnt get how to find $\sigma$ using $\frac1{\sigma\sqrt{2\pi}}$. Would you clarify further? – Nour Dec 30 '18 at 9:32
• Can you solve $\sigma$ on base of: $0.997356=\frac1{\sigma\sqrt{2\pi}}$? – drhab Dec 30 '18 at 9:36
• $\sigma = \frac1{0.997356\sqrt{2\pi}}$ which equals to 0.3999. i am not getting positive response for this answer, anything wrong i did? – Nour Dec 31 '18 at 4:33
• I cannot find a mistake in what you did and have the same outcome: $0.4$ – drhab Dec 31 '18 at 8:23