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The mileage that an electric vehicle (EV) can run before its battery wears out follows the normal distribution with mu (hu)=100,000 miles, and standard deviation 20,000 miles. The number of cars sold is one million. Questions: (1) Present or show the probability density function of battery lifetime? (2) Estimate the probability that the battery is still good after running 150,000 miles? (3) Given 90% confidence level, at least how many miles does the EV's battery can operate? (4) Assuming the EV manufacturer provides free replacements if a battery fails prior to 80,000 miles. The cost of a battery is $20,000 with capacity of 100 kWh. How much cost the EV manufacturer will incur due to the free replacement cost. (5) Among the 1 million cars sold, if the manufacturer expects to see only 10% of cars fail at the end of 100,000 miles, then what is the minimum required mean lifetime of the battery given the same standard deviation? The mileage that an electric vehicle (EV) can run before its battery wears out follows the normal distribution with mu (hu)=100,000 miles, and standard deviation 20,000 miles. The number of cars sold is one million. Questions: (1) Present or show the probability density function of battery lifetime? (2) Estimate the probability that the battery is still good after running 150,000 miles? (3) Given 90% confidence level, at least how many miles does the EV's battery can operate? (4) Assuming the EV manufacturer provides free replacements if a battery fails prior to 80,000 miles. The cost of a battery is $20,000 with capacity of 100 kWh. How much cost the EV manufacturer will incur due to the free replacement cost. (5) Among the 1 million cars sold, if the manufacturer expects to see only 10% of cars fail at the end of 100,000 miles, then what is the minimum required mean lifetime of the battery given the same standard deviation