Probability/ using R studio. Please give specific explanations for all questions thank you.

3. The life time X ofa mechanical component. cosngill]. is random. It is modelledusing an exponential distribution with a mean of 4.4 years. - If the component fas during the first year. the manufacturer agrees to give a full refund. 1- If the component fails during the secondjrear, the manufacturer agrees to give a 50% refund. 1* If the component fails after the second year. but before the fth year the manufacturer agrees to give a \"1% refund. (a) What is the probability that the component lasts more than 1 year? Give an exact answer. (4 marks) (b) What is the probability that the component lasts between 2 years and 5 years? Give an exact (4 marks) answer. (c) A particular component has already lasted 1 year. What is the probability that it will last at least (4 marks) 5 years, given it has already lasted 1 year? Give an exact answer. (d) Use R to plot the probability density function of X over the range of 0 to 50 years. (4 marks) (e) If the manufacturer sells one component, what should they expect to pay in refunds? (4 marks) (f) If the manufacturer sells 1000 components, what should they expect to pay in refunds? (2 marks) (g) Verify your answers to the parts above using simulation with rexp (). Use at least 10" simulations. (10 marks) Provide a brief explanation comparing your simulated answers with your theoretical ones. (h) The company introduces a backup system into the component, which is otherwise unchanged. (10 (bonus)) The backup system fails at a time Y modelled as an exponential distribution with mean 2.3 years. The new component fails if both the original component and the backup system fail; X and Y are independent. We are interested in the failure time f' of the new component. For full bonus marks, either. specify the CDF for t' or use numerical simulation to find E(t') (don't let me stop you from trying both if really want to)