A certain alloy is formed by combining the melted mixture of two metals. The resulting alloy contains a certain proportion of lead, say X, which may be considered a random variable.

Suppose X has the pdf, f(x) = 12x(1 x)², 0 < x < 1, and 0 elsewhere.

(a) Find the mean and variance of X.

(b) Find the cumulative distribution function of X.

(c) Suppose that $P, the net profit realized in selling this alloy (per kg) is

the following function of X : P = a+bX +cX², where a, b and c are constants. Compute the expected profit (per kg).