Answer to Question #219049 in Statistics and Probability for stan

Question

Let Z~N (0,1).Find a constant c for which P(-c≤ Z ≤ c) = 0.8664

Answer 1

By condition, the mathematical expectation is $a=0$ and the standard deviation is $\sigma = 1$ .

Then

$P( - c \le Z \le c) = 0.8664 \Rightarrow \Phi \left( {\frac{{c - a}}{\sigma }} \right) - \Phi \left( {\frac{{ - c - a}}{\sigma }} \right) = 0.8664 \Rightarrow \Phi \left( c \right) - \Phi \left( { - c} \right) = 0.8664 \Rightarrow 2\Phi \left( c \right) = 0.8664 \Rightarrow \Phi \left( c \right) = 0.4332 \Rightarrow c = 1.5$

Answer: $c=1.5$