6 The continuous random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\).
- Each of the three following sets of probabilities is impossible. Give a reason in each case why the probabilities cannot both be correct. (You should not attempt to find \(\mu\) or \(\sigma\).)
(a) \(\mathrm { P } ( X > 50 ) = 0.7\) and \(\mathrm { P } ( X < 50 ) = 0.2\)
(b) \(\mathrm { P } ( X > 50 ) = 0.7\) and \(\mathrm { P } ( X > 70 ) = 0.8\)
(c) \(\quad \mathrm { P } ( X > 50 ) = 0.3\) and \(\mathrm { P } ( X < 70 ) = 0.3\) - Given that \(\mathrm { P } ( X > 50 ) = 0.7\) and \(\mathrm { P } ( X < 70 ) = 0.7\), find the values of \(\mu\) and \(\sigma\).