Please use Rstudio if required and give specific explanations thank you.

3. The life time X ofa mechanical mmponent. tasting. is random. It is modelledusing an exponential distribution widi a mean of 4.4 years. - If the component fails during the rst year. the manufaemrer agrees to give a full refund. 1- If the component fails during the second year, the manufacturer agrees to give a E'a refund. I If the component fails after the second year. but before the fth year the manufacturer agrees to give a IU'N: refund. [El [bl What is the probability that the component lasts more dian 1 year? Give an exact answer. [:1- Haiti-is} What is the probability that the component lasts between 2 years and 5 years? Give an exact [a1- Haiti-h} answer. [c] Aparticular component has already lasted 1 year. What is the probability that it will last at least [4 111::de id} {El {fl {33 {hi 5 years. given it has alreadylasted lyear? Give an exact answer. Use R to plot the probability density function of I over the range ot' to SI] years. [4 marks] If the manufacturer sells one component, what should they expect to pay in refunds? [4 marks] If the manufacturer sells ll} components. what should they expect to pay in refunds? [2 curls} Herifyyunr answers to the parts above using simulation with rexpfl. Use atleast 11)5 simulations. [ID 111::de Provide a brief explanan'on comparing your simulated answers with your theoretical ones. The company introduces a backup system into the component, which is otherwise unchanged. [ID [bonus]} The backup system fails at a time F modelled as an exponential distribution with mean 2.3 years. The new component fails if both the original component and the backup system fail; I and Y are independent. We are interested in the failure time t' of the new component. For full bonus marks. eirei': specify the CDF for r' oruse numerical simulation to nd E{t'} [don't let me stop you from trying both if really want to}