Question: A candy company distributes boxes of chocolates with a mixture of creams, toffees, and cordials. Suppose that the weight of each box is 1 kilogram,

A candy company distributes boxes of chocolates with a mixture of creams, toffees, and cordials.

Suppose that the weight of each box is 1 kilogram, but the individual weights of the creams, toffees, and cordials vary from box to box. For a randomly selected box, let X and Y represent the weights of the creams and the toffees, respectively, and suppose that the joint density function of these variables is f(x, y) = 24xy, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, x + y ≤ 1, 0, elsewhere.

(a) Find the probability that in a given box the cordials account for more than 1/2 of the weight.

(b) Find the marginal density for the weight of the creams.

(c) Find the probability that the weight of the toffees in a box is less than 1/8 of a kilogram if it is known that creams constitute 3/4 of the weight.

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