a. For f(x1, x2, x3) as given in Example 5.10, compute the joint marginal density functi...

a. For f(x1, x2, x3) as given in Example 5.10, compute the joint marginal density function of X₁ and X₃ alone (by integrating over x2).

b. What is the probability that rocks of types 1 and 3 together make up at most 50% of the sample? [Hint: Use the result of part (a).]

c. Compute the marginal pdf of X1 alone. [Hint: Use the result of part (a).]

Reference example 5.10

When a certain method is used to collect a fixed volume of rock samples in a region, there are four resulting rock types. Let X1, X2, and X3 denote the proportion by volume of rock types 1, 2, and 3 in a randomly selected sample (the proportion of rock type 4 is 1 – X1 – X2 –X3, so a variable X4 would be redundant). If the joint pdf of X1, X2, X3 is

The notion of independence of more than two random variables is similar to the notion of independence of more than two events.