With reference to Exercise 101, find

(a) the marginal density of P;

(b) the conditional density of S given P = p;

(c) the probability that sales will be less than 30,000 units when p = 25 cents.

Exercise 101

Suppose that P, the price of a certain commodity (in dollars), and S, its total sales (in 10,000 units), are random variables whose joint probability distribution can be approximated closely with the joint probability density

Find the probabilities that

(a) the price will be less than 30 cents and sales will exceed 20,000 units;

(b) the price will be between 25 cents and 30 cents and sales will be less than 10,000 units.