N2010 P2 Q9

In this question you should state clearly the values of the parameters of any normal distribution you use.

Over a three-month period Ken makes \(X\) minutes of peak-rate telephone calls and \(Y\) minutes of cheap-rate calls. \(X\) and \(Y\) are independent random variables with the distributions \(\mathrm{N}\left( 180,{{30}^{2}} \right)\) and \(\mathrm{N}\left( 400,{{60}^{2}} \right)\) respectively.

1. Find the probability that, over a three-month period, the number of minutes of cheap-rate calls made by Ken is more than twice the number of minutes of peak-rate calls.[4]

Peak-rate calls cost \(\$0.12\) per minute and cheap-rate calls cost \(\$0.05\) per minute.

2. Find the probability that, over a three-month period, the total cost of Ken’s calls is greater than \(\$45\).[3]
3. Find the probability that the total cost of Ken’s peak-rate calls over two independent three-month periods is greater than \(\$45\).[3]