A stock solution of hydrochloric acid (HCl) supplied by a certain vendor contains small amounts of several impurities, including copper and nickel. Let X denote the amount of copper and let Y denote the amount of nickel, in parts per ten million, in a randomly selected bottle of solution. Assume that the joint probability density function of X and Y is given by

a. Find the value of the constant c so that f (x,y) is a joint density function.

b. Compute the marginal density function fX (x).

c. Compute the conditional density function fY |X (y | x).

d. Compute the conditional expectation E(Y | X = 0.4).

e. Are X and Y independent? Explain.

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