The percentage concentration, in alcohol, of a medical substance is a continuous variable X with density function f (x) = cx(1 − x)n, 0 < x < 1, where c is a real constant and n is a positive integer.

(i) Find the value of

c, as a function of n.

(ii) For n = 7, show that the average content in alcohol for this substance is 20%.

(iii) For n = 7, obtain the distribution function of X and then find each of the probabilities P(0 < X ≤ 0.25), P(0.25 < X ≤ 0.5), P(0.5 < X ≤ 0.75).

(iv) When the percentage concentration is between 0.25(j − 1) and 0.25j, the substance is sold at a price aj dollars per liter, while the cost of production is bj per liter, for j = 1, 2, 3, 4. Calculate, in terms of n, aj, and bj, the average profit made by the sale of 1 l from this substance.