In a city, homeowners use two types of heaters. The heaters of the first type constitute 30%, and their lifetimes are exponentially distributed with a mean of 10 years. The heaters of the second type are cheaper, and their lifetime is also exponential but the mean time is 8 years.
Thus, the distribution of the lifetime of a new heater that has been just bought is a mixture of exponential distributions.

(a) Is the same true for a heater chosen at random from those whose “age” is five years?

(b) Find the distribution of the remaining lifetime of such a heater. Interpret your results from a common-sense point of view.