a. The distribution of the amount of gravel (in tons) sold by a particular construction supply company in a given week is a continuous random variable X with pdf

f(x) = 3/2 (1 - x²) 0 x 1 0 otherwise

i. Determine the cumulative distribution function (cdf) of X ii. Calculate the mean of X iii. Calculate the variance of X

b. i. Prove that the mean and variance of a uniform distribution are respectively given by E(X) = (a + b)/2 and Var (X) = (b - a) ²/12, where a and b are respectively the lower and upper limits of the distribution.