The random variable X measures the concentration of ethanol in a chemical solution, and the random variable Y measures the acidity of the solution. They have a joint probability density function
f(x, y) = A(20-x-2y)
for 0 < x < 5 and 0 < y < 5, and f(x, y) = 0 elsewhere.
(a) What is the value of A?
(b) What is P(l < X < 2. 2 < Y < 3)?
(c) Construct the marginal probability density functions fX(x) and fY(y).
(d) Are the ethanol concentration and the acidity independent?
(e) What are the expectation and the variance of the ethanol concentration?
(f) What are the expectation and the variance of the acidity?
(g) If the ethanol concentration is 3, what is the conditional probability density function of the acidity?
(h) What is the covariance between the ethanol concentration and the acidity?
(i) What is the correlation between the ethanol concentration and the acidity?