The lifetime, in hundreds of hours, of a light bulb is a continuous random variable with density function f (x) = ????2xe−????x, x > 0, where ???? > 0.

(i) Calculate the distribution function of X.
(ii) What is the expected value and the variance for the lifetime of a bulb?
(iii) From past data, it has been estimated that the average length of lifetime for a bulb of this type is 2500 hours. Which value of ???? should be used to describe the distribution of bulb lifetimes? Using this value of ????, calculate the probability that a randomly selected light bulb will last for more than 2500 hours.