2. A continuous random variable X has probability density function f, where () = 5 8 (4 4 ), 1 < < 1

(I) Check that this f is a valid probability density function.

(II) Calculate ( > 1 2 )

(III) Calculate ()and ()

3. A garage is supplied with petrol every Monday morning and its weekly sales in thousands of liters is a random variable X whose probability density function f is given by () = 3 125 (5 ) 2 , 0 < < 5

(i) Find the probability that the garage's sales in a week will be less than 3000 liters.

(ii) Compute the probability that the garage will not be able to meet the demand in a week given that the capacity of its supply tank is 4000 liters.