A manufacturer of an electronic control system studies the reliability of the system by running

the system and recording the times $($in hours$)$ between breakdowns of the system. It is

proposed that the times between breakdowns for the system can be modelled by an

exponential distribution with parameter $.$ However, the times between breakdowns $(Y)$ of

the system is only recorded when the system runs for at least one hour after a breakdown.

The resulting records were given in the table below.

$($a$)$ Work out the likelihood function for this sample of recorded times between

breakdowns and calculate the maximum likelihood estimate $($MLE$)$ of the

parameter $.$ You must show all your working. Using this, deduce the MLE's

of $E(Y)$ and $V(Y).$

$($b$)$ Construct a $95\%$ confidence interval for your MLE of $$ and comment on

results.

$($c$)$ Calculate the probability of the times between breakdowns $(Y)$ of the system

being larger than $400$ hours.