Marks for each part of each question are indicated in parentheses in the questions below. Please see course outline for Part $2$ Marking Scheme for penalties.

Total marks: $10.$ This assignment is worth $5\%$ of the course mark.

Package Sorting in a Warehouse

When a shipment of packages arrives at a warehouse, an automated sorting system sorts the packages based on their destination. Sorting the packages efficiently ensures that they can be loaded onto the appropriate delivery trucks on time. It is crucial that the sorting process is completed before the trucks leave the warehouse.

$($a$)(3$ marks$)$ The time taken to load the packages onto the trucks is $5\mathrm{.}8$ hours. Sorting System $($Type A$)$ sorts one half of the shipment first and then the other half. The sorting time for each half is Normally distributed with a mean of $2\mathrm{.}5$ hours and a standard deviation of $0\mathrm{.}5$ hours. The sorting times for the two halves are correlated with a correlation coefficient of $0\mathrm{.}6.$ What is the probability that the sorting system will finish sorting the entire shipment before the trucks are fully loaded?

$($b$)(3$ marks$)$ The time taken to load the packages onto the trucks is Normally distributed with a mean of $5\mathrm{.}8$ hours and a standard deviation of $1\mathrm{.}2$ hours. Sorting System $($Type B$)$ sorts the entire shipment in a time that is Normally distributed with a mean of $4\mathrm{.}5$ hours and a standard deviation of $0\mathrm{.}9$ hours. What is the probability that the sorting system will finish sorting the shipment before the trucks are fully loaded?

When a small shipment arrives at the warehouse, it needs to be sorted quickly before being loaded onto the trucks, which takes $3\mathrm{.}2$ hours. Two sorting systems are used to sort the packages, one for each half of the shipment.

$($c$)(2$ marks$)$ Each sorting system $($Type C$)$ has a sorting time that is Normally distributed with a mean of $2\mathrm{.}8$ hours and a standard deviation of $0\mathrm{.}4$ hours, and these times are independent of each other. What is the probability that both halves of the shipment will be sorted in $3\mathrm{.}2$ hours, ready to be loaded onto the trucks?

$($d$)(2$ marks$)$ Each sorting system $($Type D$)$ has a sorting time that is Exponentially distributed with a mean of $2\mathrm{.}8$ hours, and these times are independent of each other. What is the probability that both halves of the shipment will be sorted in $3\mathrm{.}2$ hours, ready to be loaded onto the trucks?