1. ## normal distribution questions

1.) It is given that X~N( μ, σ2) and P(X<3)=P(X>7). Write down the value of μ. It is also given that 2P(X<2)=P(X<8) Find σ2.

2.)John goes to school by bus. His arrival time at the bus interchange is normally distributed with mean 7.00am and standard deviation 2 minutes. Buses are scheduled to leave promptly every 5 minutes from 6.31am.
i) if john always get on the first bus that arrives, find the probabilities that he catches the bus at a) 6.56am, b) 7.01am, c) 7.06am

ii) john will be late for school if he catches a bus after 7.06am. Assuming that the standard deviation of the arrival time remains the same, determine the new mean arrival time that will ensure that, on average, he is not late for school for more than one day in a five-day week.

1.) It is given that X~N( μ, σ2) and P(X<3)=P(X<7). Write down the value of μ. It is also given that 2P(X<2)=P(X<8) Find σ2.

[snip]
5 is halfway between 3 and 7 ......

$2 \Pr\left( z < - \frac{3}{\sigma}\right) = \Pr\left( z < \frac{3}{\sigma}\right)$

$\Rightarrow 2 \Pr\left( z > \frac{3}{\sigma}\right) = \Pr\left( z < \frac{3}{\sigma}\right)$

$\Rightarrow 2 \left[1 - \Pr\left( z < \frac{3}{\sigma}\right) \right]= \Pr\left( z < \frac{3}{\sigma}\right)$

$\Rightarrow 2 = 3 \Pr\left( z < \frac{3}{\sigma}\right)$

$\Rightarrow \Pr\left( z < \frac{3}{\sigma}\right) = \frac{2}{3}$.

Therefore $\frac{3}{\sigma} = \, ....$ (using tables in reverse, an inverse normal problem).

Now solve for $\sigma$.