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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?

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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 $P(X\leq0)$.

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  • $\begingroup$ 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? $\endgroup$ – Nour Dec 30 '18 at 9:32
  • $\begingroup$ Can you solve $\sigma$ on base of: $0.997356=\frac1{\sigma\sqrt{2\pi}}$? $\endgroup$ – drhab Dec 30 '18 at 9:36
  • $\begingroup$ $\sigma = \frac1{0.997356\sqrt{2\pi}}$ which equals to 0.3999. i am not getting positive response for this answer, anything wrong i did? $\endgroup$ – Nour Dec 31 '18 at 4:33
  • $\begingroup$ I cannot find a mistake in what you did and have the same outcome: $0.4$ $\endgroup$ – drhab Dec 31 '18 at 8:23

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